Study On Periodic Solutions And Their Stabilities Of Eccentric Rotors Under Unbalanced Magnetic Excitation

In the modern manufacturing and life, the motor plays an important role and is wide used, therefore, the study of their dynamic characteristics are particularly important. We adopt the method of constructing analytical solutions to explore their dynamic characteristics. For simulating the three-phase synchronous motor system, we consider a rotor system which has 4-DOF and contains the effects of rotary inertia, gyroscopic moments, the resistance outside the shaft, eccentric mass and the unbalanced magnetic pull. And the dynamics equation of this system isIn this equation m is the quality of the disk, e1 is the eccentricity, Ω is spin velocity, Jd is the moment of inertia about its diameter,k11,k33,k14,c11,c33 is elastic coefficients and damping coefficients respectively, PxUMP, PyUMP are the unbalanced magnetic pull in x, y directions respectively, can approximate as sum of several sine and cosine motivations whose amplitude is polynomial with variables x, y and frequency is double electrical frequency.In order to construct its periodic solution, we make the non-autonomous system of differential equations of motion with parametrically exciting force be transformed to an autonomous one by introducing a rotating coordinate frame mz1+(c11+imΩ)z1+(k11-mΩ2+ic11Ω)z1-ik14z2= g(z1) Jdz2+(c33+i2JdΩ-iH)z2+(k33-JdΩ2+ic33Ω+HΩ)z2+ik14z1= 0 (2) here, z1=(x+iy)e-iΩt, z2= (θx+iθy)e-iΩt,g(z1) are polynomial with variables z1.Then the circular whirling of the model rotor are converted into equilibrium solutions of the autonomous system by converting second order differential equation with 4 variables to first order differential equation with 8 variables. here, g1 (x, y), g2 (x, y) are polynomial with variables x, y.Let right side of the equation (3) is zero, thus we can obtain the amplitude of transverse vibration and amplitude of rotation angle by solving a system of polynomial equations with 4 unknowns only. But when more than one solution is possible, the system may settle in any of them, depending on the stability of equilibrium solution.In this paper,2 methods are being used to identify the stability of periodic solutions: One way is according to the Lyapunov stability criterion, which is the stability of periodic solutions is determined by positive or negative of all the eigenvalue real part of Jacobin matrix of equation(3); The other method is given under symmetric load condition, the solution of equations can be written in form of solution and its small disturbance, and through the transformation and linearization of small disturbance, then periodic variable coefficient linear is obtained, and then based on the Floquet theory to judge differential equation, which is the stability of periodic solution.The research has indicated that the periodic solution constructed and the identification of its stability are correct. Furthermore, the unbalanced magnetic pull usually cause larger amplitude of circular whirling and rotation angle; meanwhile, radius of circular whirling and bending vibration angle becomes smaller with the increase of pole-pair number; radius of circular whirling and rotation angle is bigger without the unbalanced magnetic pull considered;2-DOF model is inaccurate to simulate the rotor system that the disc is not in middle of two fulcrums; the position of the rotor continuously influence on its vibration, the more rotor is close to the midpoint of two fulcrums, the more its transverse vibration is intense, but the amplitude of bending vibration increases initially and decreases afterwards. When the rotor’s position close to midpoint, the amplitude is zero.