Abstract
In order to reveal synchronization characteristics of a weakly damped system with two rotors mounted on different vibrating bodies, we propose a simplified physical model. Vibration of the system is discussed by the average method, which can separate fast motions (high frequency) from slow motions (low frequency). Theoretical research shows that vibration torque is the key factor to balance the energy distribution between rotors. For the system with rotational frequencies larger than the nature frequencies, the coupling characteristic frequency or characteristic frequency curve should be considered. As the coupling frequency is close to the characteristic frequency, or the vibration state is close to the characteristic frequency curve, selfsynchronization of two rotors can be obtained easily.
1. Introduction
The socalled selfsynchronization phenomenon corresponds to the consistency or certain relationship of systems’ parameters caused by their internal couplings and has been widely involved in nonlinear vibration, hydraulic [13], electromechanical coupling, automatic control theory and other fields [48].
Huygens was the first person who observed the synchronization of pendulum clocks in the 17th Century. In [4, 5], Czolczynski et al. presented different synchronous behaves of two or $n$ pendula installed on a frame. The selfsynchronization theory of rotors was developed by Bleckman [1, 2] with averaging method in the middle of the 20th century. Wen and Zhao et al. [7] modified the averaging method and proposed the average method with two small parameters. Zhang [8] deduced the synchronization condition and the synchronization stability for the vibrating system with three rotors. Hou and Fang [9, 10] investigated a vibrating screen based on the model of a rotorpendulum system.
The above researches are mostly focused on the synchronization of pendula or rotors installed on the same vibrating frame. In this paper, we propose a vibrating system with two rotors mounted on two different vibrating bodies.
2. Dynamical equations of the vibrating system
As shown in Fig. 1, two rotors are mounted on different vibrating bodies. The vibrating body (${M}_{i}$) ($i=$ 1, 2) can move in horizontal direction (${x}_{i}$) and is installed on the foundation by the spring. The two bodies are connected by a coupling spring. Counterclockwise direction is taken as positive. Inertia moment and eccentricity of the rotor on its mass center are given by ${j}_{i}$ and ${r}_{i}$. Other variables are show in Fig. 1. In this paper, synchronization of rotors is analyzed in a nonresonant vibrating system, in which rotation frequencies of rotors are larger than nature frequencies of vibrating bodies. The system is denoted as afterresonance system. We assume that ${\dot{\phi}}_{1}$, ${\dot{\phi}}_{2}>\text{2}{\omega}_{1}$, $\text{2}{\omega}_{2}$, where ${\omega}_{1}=\sqrt{{k}_{1}/{M}_{1}}$, ${\omega}_{2}=\sqrt{{k}_{2}/{M}_{2}}$.
Fig. 1Simplified model of the system
The electromagnetic torque and resistance moment of the driving motor$i$ are ${T}_{ei}$ and ${T}_{fi}$ (${T}_{fi}={f}_{ri}\bullet {\dot{\phi}}_{i}$) respectively. When ${\dot{\phi}}_{i}$ ($i=$ 1, 2) fluctuates near the frequency ${\omega}_{n}$, the influence of electromagnetic leakage can be neglected, and the driving force of induction motor can be linearized as ${T}_{ei}=n{L}_{mi}^{2}{U}_{0}^{2}\left({\omega}_{si}n{\omega}_{n}\right)/\left({{L}_{si}^{2}{\omega}_{si}{R}_{ri}\omega}_{si}\right)$, where ${n}_{i}$, ${L}_{mi}$, ${\omega}_{si}$, ${L}_{si}$ and ${R}_{ri}$ ($i=$1, 2) correspond to the pole number, mutual inductance, synchronous speed, stator inductance and rotor resistance of the motor; ${U}_{0}$ is the voltage amplitude. As selfsynchronization of rotors is achieved, speed fluctuations of rotors are small [1, 2]. Therefore, small variables can be neglected. Introducing ${\omega}_{p}=\sqrt{{k}_{p}/{M}_{1}}$, $\eta ={M}_{2}/{M}_{1}$, ${J}_{1}={j}_{1}+{m}_{1}{r}_{1}^{2}$ and ${J}_{2}={j}_{2}+{m}_{2}{r}_{2}^{2}$, we have:
${\ddot{x}}_{2}+{\omega}_{2}^{2}{x}_{2}+\eta {\omega}_{p}^{2}\left({x}_{2}{x}_{1}\right)=\frac{{m}_{2}}{{M}_{2}}\delta {r}_{2}{\dot{\phi}}_{2}^{2}\mathrm{cos}{\phi}_{2},$
${J}_{1}{\ddot{\phi}}_{1}={T}_{e1}{T}_{f1}+{m}_{1}{r}_{1}{\ddot{x}}_{1}\mathrm{sin}{\phi}_{1},$
${J}_{2}{\ddot{\phi}}_{2}={T}_{e2}{T}_{f2}+\delta {m}_{2}{r}_{2}{\ddot{x}}_{2}\mathrm{sin}{\phi}_{2}.$
The synchronous speed of two rotors is denoted by ${\omega}_{n}$. When selfsynchronization of rotors is achieved [1, 3], phases of the two rotors can be denoted as ${\phi}_{1}={\omega}_{n}t+{\alpha}_{1}$, ${\phi}_{2}={\omega}_{n}t+{\alpha}_{2}$, where ${\alpha}_{1}$ and ${\alpha}_{2}$ are slowlyvarying parameters. From Eq. (1), we obtain:
${x}_{2}={\mu}_{21}\mathrm{cos}{\phi}_{1}+{\mu}_{22}\mathrm{cos}{\phi}_{2},$
where:
${\mu}_{21}=\frac{{m}_{1}{r}_{1}{\omega}_{n}^{2}{\omega}_{p}^{2}}{N},{\mu}_{22}=\frac{\delta {m}_{2}{r}_{2}{\omega}_{n}^{2}\left({\omega}_{1}^{2}+\eta {\omega}_{p}^{2}{\omega}_{n}^{2}\right)}{\left(\eta N\right)},$
$N={M}_{1}\left[\left({\omega}_{1}^{2}+\eta {\omega}_{p}^{2}{\omega}_{n}^{2}\right)\left({\omega}_{2}^{2}+{\omega}_{p}^{2}{\omega}_{n}^{2}\right)\eta {\omega}_{p}^{4}\right].$
${\mu}_{11}$, ${\mu}_{12}$, ${\mu}_{21}$ and ${\mu}_{22}$ show the coupling effects in the system. Rotors are driven by motors, and the resistance is approximately proportional to its speed. The average values of resultant torques of rotors are denoted by ${P}_{1}$, ${P}_{2}$. As the system is stable, we have:
${P}_{2}={\int}_{t}^{t+T}{T}_{e2}{T}_{f2}+\delta {m}_{2}{r}_{2}{\ddot{x}}_{2}\mathrm{sin}{\phi}_{2}dt={T}_{e2}\left({\omega}_{n}\right){T}_{f2}\left({\omega}_{n}\right)+\frac{1}{2}{T}_{v},$
where ${T}_{v}=\delta {m}_{2}{r}_{2}{\mu}_{21}{\omega}_{n}^{2}\mathrm{sin}\left({\alpha}_{1}{\alpha}_{2}\right)={m}_{1}{r}_{1}{\mu}_{12}{\omega}_{n}^{2}\mathrm{sin}\left({\alpha}_{1}{\alpha}_{2}\right)$ is the vibration torque (VT). Introducing the variable substitutions:
$\mathrm{sin}\u2206\alpha =\frac{\left[\u2206{T}_{e}\left({\omega}_{n}\right)\u2206{T}_{f}\left({\omega}_{n}\right)\right]}{\left({m}_{1}{r}_{1}{\mu}_{12}{\omega}_{n}^{2}\right)},$
can be obtained. Thus, the synchronization condition can be expressed as follows [1]:
The stability criterion of the synchronous state can be discussed based on Lyapunov stability theory, it can be deduced as:
3. Discussions of theoretical results
In this paper, two rotors rotate in the same direction, that is, $\delta =$ 1. In this paper, ${L}_{mi}=$ 0.14 H, ${L}_{si}=$ 0.12 H, ${R}_{ri}=$ 0.6 $\mathrm{\Omega}$, ${\omega}_{si}=$ 314 rad/s, ${U}_{0}=$ 220 V, $n=$ 2 and other parameters of the system are shown in Table 1.
Table 1Parameters of the system
Parameters  ${M}_{i}$ [kg]  ${m}_{i}$ [kg]  ${j}_{i}$ [kg·m^{2}]  ${r}_{i}$ [m]  ${f}_{i}$ [N·s/m]  ${k}_{i}$ [N/m]  ${f}_{ri}$ [N·m·s/rad] 
Rotor 1  300  3.5  0.3  0.15  200  7.5×10^{5}  3×10^{2} 
Rotor 2  200  2.5  0.3  0.1  200  7.4×10^{5}  1.47×10^{1} 
We take ${T}_{v\mathrm{m}\mathrm{a}\mathrm{x}}\left({\omega}_{p}^{2}\right)=\delta {m}_{1}{r}_{1}{m}_{2}{r}_{2}{\omega}_{n}^{4}{\omega}_{p}^{2}/N$ into consideration. ${T}_{v}=\mathrm{sin}\u2206\alpha {T}_{v\mathrm{m}\mathrm{a}\mathrm{x}}\left({\omega}_{p}^{2}\right)$ can be obtained easily. Thus, ${T}_{v\mathrm{m}\mathrm{a}\mathrm{x}}\left({\omega}_{p}^{2}\right)$ is the maximum vibration torque of the system (MVT). For the afterresonance system, denominator of $N$ may be zero when ${\omega}_{p}$ come to be a specific value $\chi $. ${\chi}^{2}$ is deduced as:
$\chi $ is called the characteristic frequency (CF) of the system, as shown in Fig. 2(a) and Fig. 3. In the coordinates of ${\omega}_{n}$ and ${\omega}_{p}^{2}$, the characteristic curve composed of characteristic frequencies at different synchronous speeds is shown in Fig. 2(b). The curve of ${T}_{v\mathrm{m}\mathrm{a}\mathrm{x}}\left({\omega}_{p}^{2}\right)$ accompanied with ${\omega}_{p}^{2}$ is shown in Fig. 3 when ${\omega}_{n}$ is a certain value (for example, ${\omega}_{n}=$ 155 rad/s). The synchronous speed varies with the stiffness of the coupling spring and it can be obtained by numerical simulation.
When ${\omega}_{p}^{2}$ approach ${\chi}^{2}$ from the left side, ${T}_{v\mathrm{m}\mathrm{a}\mathrm{x}}$ tends to infinity; when ${\omega}_{p}^{2}$ continues to increase over ${\chi}^{2}$ to infinity, ${T}_{v\mathrm{m}\mathrm{a}\mathrm{x}}$ gradually decreases and tends to a constant value. In Fig. 3, the curve ${T}_{v\mathrm{m}\mathrm{a}\mathrm{x}}$ is divided into four parts by the curves of $\pm \left(\u2206{T}_{e}\left({\omega}_{n}\right)\u2206{T}_{f}\left({\omega}_{n}\right)\right)$. The four parts are denoted by LA (passing through point A), LB (passing through point B), LC (passing through point C) and LD (passing through point D) respectively. According to Eq. (4), selfsynchronization of two rotors cannot be obtained when system state occurs on LA and LD; On the contrary, rotations of two rotors can be selfsynchronizing when system state occurs on LB and LC.
As coupling frequency ${\omega}_{p}$ is close to the characteristic frequency $\chi $, or the system state is near the characteristic frequency curve, the system coupling performance is strong, and the self synchronization of two rotors can be obtained easily. And it is convenient to control the synchronization performance by adjusting the coupling spring stiffness ${k}_{p}$.
Fig. 2Surface of the maximum vibration momentvarying with the coupling frequency and synchronous speed in the afterresonance system
a) The threedimensional diagram
b) The twodimensional diagram
Fig. 3Relationship between the maximum vibration moment and coupling frequency
4. Simulations for synchronization of two rotors
Simulations are carried out with ${\omega}_{p}^{2}$ set to be 6800 (rad/s)^{2}, 11600 (rad/s)^{2}, 16000 (rad/s)^{2}, 35600 (rad/s)^{2} and infinity respectively, corresponding to points A, B, C and D.
Fig. 4 shows the simulation results with ${\omega}_{p}^{2}=$ 6800 (rad/s)^{2}. From Figs. 2 and 3, we can suggest that rotations of two rotors would not be selfsynchronizing in this case. As shown in Fig. 4, speeds of the two rotors are not consistent. Fig. 5 shows the simulation results with ${\omega}_{p}^{2}=$ 11600 (rad/s)^{2}. In this case, $\left{T}_{v\mathrm{m}\mathrm{a}\mathrm{x}}\right>\left{\u2206T}_{e}\left({\omega}_{n}\right){\u2206T}_{f}\left({\omega}_{n}\right)\right$. $\u2206\alpha $ is calculated to be 3.28 rad. As shown in Fig. 5, speeds of the two rotors reach the same value around 1.5 s, and $\u2206\alpha $ is stable at 15.83 rad (15.83 – 4 ×$\pi $ = 3.26 (rad)). Fig. 6 shows the simulation results with ${\omega}_{p}^{2}=$ 16000 (rad/s)^{2}. Parameters of system satisfy the selfsynchronization conditions in this case too. $\u2206\alpha $ is calculated to be 3.65 rad. As shown in Fig. 6, speeds of the two rotors reach the same value around 1.5 s, and $\u2206\alpha $ is stable at 16.21 rad (16.21 – 4 × $\pi $ = 3.64 (rad)). The numerical results in Figs. 5 and 6 are consistent with the theoretical analysis.
Fig. 4Simulation results of the afterresonance system when the coupling stiffness is 1.36×106 N/m
a) Rotational speeds of two rotors
b) The phase difference between two rotors
Fig. 5Simulation results of the afterresonance system when the coupling stiffness is 2.32×106 N/m
a) Rotational speeds of two rotors
b) The phase difference between two rotors
Fig. 6Simulation results of the afterresonance system when the coupling stiffness is 3.20×106 N/m
a) Rotational speeds of two rotors
b) The phase difference between two rotors
Fig. 7Simulation results of the afterresonance system when the coupling stiffness is 7.12×106 N/m
a) Rotational speeds of two rotors
b) The phase difference between two rotors
Fig. 8Simulation results of the afterresonance system when the coupling stiffness is tending to infinity
a) Rotational speeds of two rotors
b) The phase difference between two rotors
Figs. 7 and 8 show the simulation results when ${\omega}_{p}^{2}$ is set to be 35600 (rad/s)^{2} and tend to infinity respectively. According to Figs. 2 and 3, rotations of two rotors would not be selfsynchronizing in these cases. As shown in Figs. 7(a) and 8(a), speeds of the two rotors are not consistent when the system reaches stable conditions; Figs. 7(b) and 8(b) demonstrate the asynchrony of the two rotors. Simulation results confirm the theoretical analysis.
5. Conclusions
Selfsynchronization of two rotors can be observed in a weakly damped nonresonant vibrating system with two rotors mounted on different bodies. The synchronization condition is that the vibration torque is large enough to overcome the input torque difference between two rotors. For the afterresonance system, there is a characteristic frequency or characteristic frequency curve. As the coupling frequency is close to the characteristic frequency, the coupling effects of the system can be strong, and selfsynchronization of two rotors occurs easily. While there is a big difference between the coupling frequency and the characteristic frequency, selfsynchronization will not be achieved.
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About this article
This study is supported by International Science and Technology Cooperation Program of China (Grant No. 2015DFR70660).