Synchronization of Two Non-Identical Coupled Exciters in a Non-Resonant Vibrating System of Linear Motion. Part I: Theoretical Analysis

Abstract

In this paper an analytical approach is proposed to study the feature of frequency capture of two non-identical coupled exciters in a non-resonant vibrating system. The electromagnetic torque of an induction motor in the quasi-steady-state operation is derived. With the introduction of two perturbation small parameters to average angular velocity of two exciters and their phase difference, we deduce the Equation of Frequency Capture by averaging two motion equations of two exciters over their average period. It converts the synchronization problem of two exciters into that of existence and stability of zero solution for the Equation of Frequency Capture. The conditions of implementing frequency capture and that of stabilizing synchronous operation of two motors have been derived. The concept of torque of frequency capture is proposed to physically explain the peculiarity of self-synchronization of the two exciters. An interesting conclusion is reached that the moments of inertia of the two exciters in the Equation of Frequency Capture reduce and there is a coupling moment of inertia between the two exciters. The reduction of moments of inertia and the coupling moment of inertia have an effect on the stability of synchronous operation.

Synchronization of Two Non-Identical Coupled Exciters in a Non-Resonant Vibrating System of Linear Motion. Part II: Numeric Analysis

“…Step 3: Let s 0 = s e and calculate the torque of frequency capture, T Captrue , and the residual electromagnetic torques of two motors, T Diffenrence [23], respectively.…”

“…The characteristic equation for the eigenvalue of equations of frequency capture is rewritten as the following [23] …”

“…When H 0 < 0, the synchronous operation is unstable [23]. Therefore, the stability condition of the system can be expressed as follows…”

“…In the first part of the present investigation [23], an analytical approach was proposed to study the feature of frequency capture for two non-identical coupled exciters in a non-resonant vibrating system of linear motion, which converts the problem of frequency capture and synchronization of two exciters into that of existence and stability of zero solution of Equations of Frequency Capture. The moment of inertia of each exciter in the Equations of Frequency Capture reduces and there is a coupling moment of inertia between two exciters.…”

“…The effects of W s and W c on the load torques and the moments of inertia of two exciters also depend on the phase difference between two exciters. According to the expressions of W s0 and W c0 [23] /2, the efficient of sine coupling effect, m c W s , and the cosine of phase difference of two exciters, cos 2α, which describes the torque of coupling sine effect; and the last is the product of the coupling kinetic energy, m c rω 2 m0 /2, the coefficient of cosine coupling effect, m c W c , and the sine of phase difference of two exciters, sin 2α, which expresses the torque of coupling cosine effect. The last two items stem from the coupling effect of two exciters; and their values for two motors are same.…”