We consider synchronization and stability of two unbalanced rotors reversely and fast excited by induction motors fixed on an oscillating body. We explore the energy balance of the system and show how the energy is transferred between the rotors via the oscillating body allowing the implementation of the synchronization of the two rotors. An approximate analytical analysis, energy balance method, allows deriving the synchronization condition, and the stability criterion of the synchronization is deduce by disturbance differential equations. Later, to prove the correctness of the theoretical analysis, many features of the vibrating system are computed and discussed by computer simulations. The proposed method may be useful for analyzing and understanding the mechanism of synchronization, stability, and energy balance of similar fast rotation rotors excited by induction motors in vibrating systems.
Synchronization is commonly observed to occur in nature, such as synchronous oscillations of the nuclei and cells of malignant tumors in biological objects and geosynchronous satellites rotating around the earth in celestial mechanics. All synchronous regimes arise due to natural properties of the process themselves and their natural interaction. The history of synchronization investigation goes back to the 17th century when Huygens observed weak synchronization of two pendulum clocks in a ship [
In this paper, we further extend investigation of a generalization of the vibrating system on synchronization and stability of unbalanced rotors fast excited by induction motors and show how the energy is transferred between rotors via the oscillating body allowing the implementation of synchronization of rotors. We use the type model where two rotors reversely and fast excited by the induction motors are fixed on an oscillating body in a far-resonant vibrating system (i.e., the operating frequency of the system is about 4–10 times of its natural frequency, and the damping value is very small). The performed approximate analytical analysis allows deriving the synchronization condition and stability criterion, in addition to explaining the synchronization discipline with considering diversity features of the vibrating system. Finally, some numerical simulations are performed to prove the correctness of the theoretical analysis.
This paper is organized as follows. Section
The analyzed system is shown in Figure
Dynamic model of the vibration system.
In synchronous state, multiplying (
Next multiplying (
Then multiplying (
At last, multiplying (
Adding together (
Adding (
In actual engineering applications, the same type of motors keeps the different electrical characteristics as manufacturing tolerances. According to (
We assume that the average phase angular and rotation velocity of the rotors are
As shown in Figure
According to (
In the synchronous state of the two rotors, the phase angle difference
Specifying
To ensure the existence of the solution to
The stability criterion of the synchronous state of the two rotors in the vibrating system will be discussed in the following section by solving a disturbance differential equation. Introducing disturbance parameters
From (
According to the stability theory, the coefficient of the third item of (
In our numerical verification, we consider the following values of the system parameters. We assume the parameters of the two motors are identical (i.e., rated power 0.7 Kw, rated voltage 220 V, rated frequency 50 Hz, pole pairs 2, stator resistance 0.56
Section
According to (
The stable phase difference with different features: (a)
We assume
Subsisting the above given motors’ parameters into (
Further analyses have been performed by numerical simulations. Our results have been obtained by numerical integration (by 4th order Runge-Kutta method) of (
In calculations for synchronization of the two rotors in power-supplying and power-cutting states, we consider the following values of the dynamics parameters (identical mass and symmetric location of the two rotors):
Figures
Numerical results for two identical opposite rotors.
Figures
According to (
In the power-supplying stage, the part energy supplied by the two induction motors is dissipated by their friction dampers (
In the power-cutting stage, the power source on the second rotor is cut off, and so the electromagnetic torque of the rotor is equal to zero within the synchronization state; that is,
Energy transmission. Figures (a), (b), (c), and (d) are depicted according to (
Power-supplying state
Power-cutting state
Nonidentical rotors
Asymmetric location of rotors
In calculations of the synchronization ability of the vibrating system excited by the two nonidentical rotors, we consider the following values of the dynamics parameters (nonidentical mass and symmetric location of the two rotors):
Figures
Numerical results for two no-identical opposite rotors.
In the synchronous state, the part energy supplied by the two induction motors is dissipated by their friction dampers (
To further verify the synchronous stability of the two rotors, it is necessary to perform simulations for the vibrating system with a phase disturbance on the rotors. We consider the following values of the system parameters (identical mass and asymmetric location of the two rotors):
Figures
Numerical results for the asymmetric location and disturbance of rotors.
Figures
In the synchronous state, the part energy supplied by the two induction motors is dissipated by their friction dampers (
In summary, the energy balance method is employed to extend investigation of a generalization of the vibrating system on synchronization and synchronous stability of unbalanced rotors excited by induction motors with fast rotation, on which we show how the energy is transferred between rotors via the oscillating body allowing the implementation of synchronization of rotors. In order to ensure the synchronous and stable operation of the rotors, the dynamics parameters should satisfy both the condition of synchronization and the criterion of synchronous stability. To prove the correctness of the theoretical analysis, many features of the vibrating system are computed and discussed with computer simulations. It can be found that, in the power-supplying state no matter how larger value of parameters
The authors declare that there is no conflict of interests regarding the publication of this paper.
This study is supported by National Natural Science Foundation of China (Grant no. 51074132).