Forward-backward internal resonances in asymmetrical rotors under electromagnetic and gravitational fields

This study presents a comprehensive analytical and numerical investigation of an asymmetrical rotating shaft subjected to electromagnetic loading and gravitational effects. Employing Hamilton principle and the harmonic balance method, dimensionless equations of motion and modulation equations for the gravity-induced 1:-1 internal resonance (via static deflection coupling forward and backward whirling modes), as well as simultaneous super-harmonic and 1:-1 internal resonances. Linear stability analysis yields a Campbell diagram with two critical speeds that decrease monotonically as the electromagnetic parameter increases. Backbone-curve and frequency-response analyses indicate that the electromagnetic softening effect shifts the backbone curves toward lower speeds and reduces peak amplitudes, while gravity introduces new bifurcations, right-inclined response branches, and narrow frequency ranges of simultaneous forward/backward instability. Under primary-induced internal resonance, the asymmetrical system exhibits up to five coexisting steady-state solutions (three stable), compared to three in the symmetrical system (one stable). In contrast, super-harmonic–induced internal resonance generates up to seven solutions, many of which are unstable. Analytical predictions show excellent agreement with numerical results and time-history/FFT simulations. These findings quantify the influence of electromagnetic fields, gravity, and asymmetry on the onset of complex modal interactions, offering practical guidelines for tuning rotor systems to avoid deleterious internal resonances in high-speed applications.