Synchronization of the four identical unbalanced rotors in a vibrating system of plane motion

Abstract

A new mechanism is proposed to implement the synchronization of the four unbalanced rotors in a vibrating system, which consists of a main rigid frame (MRF) and two accessorial rigid frames (ARF). An analytical approach is developed to study the coupling dynamic characteristics of the four unbalanced rotors, which converts the problem of synchronization of the four unbalanced rotors into the existence and the stability of zero solutions for the non-dimensional differential equations of the angular velocity disturbance parameters (NDDEDP). The stability of zero solutions of the NDDEDP is decomposed into that of its generalized system and a system of the three first order differential equations for the disturbance parameters of the phase differences. The coupling dynamic characteristic of the four unbalanced rotors includes the inertia coupling, the stiffness coupling of angular velocity and the load torque coupling. The non-dimensional inertia coupling matrix is symmetric, the non-dimensional matrix of the stiffness coupling of angular velocity is antisymmetric and its diagonal elements are all negative. Hence, the general system of the NDDEDP automatically satisfies the generalized Lyapunov equations when the non-dimensional inertia coupling matrix is positive definite and its elements are all positive. Using Routh-Hurwitz criterion the condition of stability of differential equations for the disturbance parameters of the phase differences is obtained. The load torque coupling makes the vibrating system have the dynamic characteristic of selecting motions and self-synchronization of the four unbalanced rotors arises from the dynamic characteristic of selecting motion of the vibrating system. When the two coefficients of coupling cosine effect of phase angles are all greater than 0 and the three indexes of synchronization are all far greater than 1, the vibrating system can implement an elliptical motion of the main rigid frame required in engineering. Numeric results show that the structural parameters of the proposed mechanism can guarantee the non-dimensional inertia matrix to be always positive definite. Computer simulation is carried out to verify the results of the theoretical investigation. Nomenclature M; M i : (i=1, 2) mass matrix of the MRF, M= diag{ , }; m m mass matrix of the ARF i, 1 2 a a diag{ , } M m m   M M ij : (i = 1, 2; j = 1, 2) mass matrix of the unbalanced rotor ij, 0 0 diag{ , } ij m m  M J; J a : moment of inertia of the MRF about its mass center; moment of inertia of the ARFs about their spin axis J 0ij :(i=1, 2; j=1, 2) moment of inertia of the motor's rotor ij r ij :(i =1, 2; j=1, 2) eccentric radius of the unbalanced rotor ij, r ij = r 0 G: mass center of the MRF o i : (i=1, 2) center of the spin-axis of the ARF i 406

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