The free and forced flexural vibrations are investigated for rotors of electric motors operating in unsymmetrical magnetic fields. The magnetic potential energy reserved in the air-gap is analytically derived and the unbalanced magnetic pull is obtained through the law of energy conservation. With this excitation, the equations of motion of the unbalanced rotor are developed for nonlinear displacements response. For small dynamic eccentricities, the equations of motion are simplified and the rotor is compared to a free Duffing oscillatory system. The analytic solution for forced vibrations subject to residual mass-unbalance excitations is also obtained. Jump phenomenon in the solution is pointed out, and the effects of initial eccentricity and flux density on the natural frequency are also investigated.
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