The model of the vibration system. (a) dynamic model of the double vibro-body system with two induction motors rotating in the same direction, (b) the reference frame system. doi:10.1371/journal.pone.0126069.g001 

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... word “ synchronization ” is often encountered in both scientific and everyday language. Our surroundings are full of synchronization phenomenon, which is considered as an adjust- ment of rhythms of oscillating objects due to their internal weak couplings. For examples: vio- linists play in unison; insects in a population emit acoustic or light pulses with a common rate; birds in a flock flap their wings simultaneously; the heart of a rapidly galloping horse contracts once per locomotory cycle. Huygens firstly described the notion of the synchronization by experiments that two pendulum clocks hung on a common support in 1665 [1]. Pol showed that the frequency of a generator can be entrained, or synchronized, by a weak external signal of a slightly different frequency in 1920 [2]. In the middle of the nineteenth century, Rayleigh [3] described the interesting phenomenon of synchronization in acoustical systems. The first En- glish monograph related to the synchronization problems is written by Blekhman [4]; he pri- marily addressed mechanical oscillators, pendulum clocks in particular, systems with rotating elements, technological equipment, but also some electronic and quantum generator; many years later, he also investigated controlled synchronization of two vibroactuators based on a speed-gradient[5, 6]. Pikovsky issued his monograph that consider synchronization as a universal concept in nonlinear sciences and review classical results on the synchronization of peri- odic oscillators [7]. Zhang investigated the synchronization problem for a class of discrete-time complex-valued neural networks with time-varying delays [8]. Nowadays, the researchers mainly focus synchronization on physical, biological, chemical and social systems, etc. In physical systems, the most representatives are synchronization of complex networks and mechanical systems. For the synchronization of complex networks, Arenas reported the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology [9]. How the feedback from dynamical clusters can shape the network connection weights and an adaptive network spontaneously forms scale- free structure were explored by Yuan Wang[10, 11]. For the synchronization of pendulum clocks, Senator developed synchronization of two coupled, similarly sized, escapement-driven pendulum clocks [12]. Jovanovic studied two models of connected pendulum clocks synchro- nizing their oscillations, a phenomenon originally observed by Huygens, with the Poincare ́ method, and they found that the in-phase linear mode damps out faster than the anti-phase mode [13]. Koluda considered two and multiple self-excited double pendula hanging from a horizontal beam with the energy balanced method, on which they found how the energy is transferred between the pendula via the oscillating beam allowing the pendula ’ synchroniza- tion[14 – 16]. For synchronization of multiple coupling rotors, Wen employed the average method to study synchronization and stability of multiple unbalanced rotors hung on a vibro- body in vibration systems, and applied such synchronization theory to invent many synchronization machines [17]. Sperling presented analytical and numerical investigations of a two- plane automatic balancing device for equilibration of rigid-rotor unbalance [18]. Balthazar ex- amined self-synchronization of four non-ideal exciters in non-linear vibration system via numerical simulations [19, 20]. Djanan explored the condition for which three motors working on a same plate can enter into synchronization with the phase difference depending on the physical characteristics of the motors and the plate, and it is indicated that one can obtain a re- duction of vibration when the motors are different and rotates in opposite directions [21]. Zhao proposed the average method of modified small parameters to investigate the synchronization of more than two exciters in far vibration systems, which immensely simplify the process for solving the theory approximation solution [22 – 28]. Later, Fang applied Zhao ’ s method to investigate the self-synchronization of two homodromy rotors coupled with a pendulum rod in a far-resonant vibration system [29]. The above-mentioned research of the mechanical systems is mainly synchronization of the pendula or the rotors directly installed on a movable beam or vibro-body. In this paper, we consider the synchronization and the synchronization stability of two homodromy unbalanced rotors installed on two vibro-bodies, respectively. The synchronization implementation of the two rotors relies on the coupling springs between vibro-body 1 and 2. The performed approximate analytical analysis, building on the original work of Zhao Chunyu, allows deriving the synchronization condition and stability criterion and explaining the synchronization discipline with considering diversity features of the vibration system. Finally, some numerical simulations are performed to verify the correctness of the theoretical analysis. This paper is organized as follows. The second section describes the considered model and dynamics equations of the vibration system. The third section we explain our method to derive the synchronization condition and the synchronization stability criterion of the system. The fourth section presents the results of our numerical simulations for the theoretical approximate solutions. The fifth section gives some computer simulations to verify our theoretical solutions. Finally, we summarize our results in the last section. Fig 1 shows the dynamics model of the considered vibration system, which consists of two rigid vibro-bodies (vibro-body 1 and vibro-body 2), on which two induction motor are installed, respectively. Each of the vibro-body is supported on an elastic foundation consisting of four springs symmetrically installed. Rigid vibro-body 1 is connected with vibro-body 2 by some stronger stiffness springs ( k x1 , k y1 , k ψ 1 ), and vibro-body 1 is connected a fix foundation with some weaker stiffness springs ( k x2 , k y2 , k ψ 2 ). The two homodromy unbalanced rotors, driven separately by two induction motors, are installed in the vibro-bodies with the equal dis- tance l from the rotation point of the rotor to the mass center of the vibro-body. During the starting process, three motors are supplied with the electric source at same time. The mass centers of the rigid vibro-body 1 and 2 are o 1 and o 2 , respectively. As illustrated in Fig 1(b), six reference frames of the system can be assigned as follows: the fixed frames o x y and o x y ; the non-rotating moving frames o 1 x 1 y 1 and o 2 x 2 y 2 , that undergoes the translation motion while remaining parallel to o 1 x 1 y 1 and o 2 x 2 y 2 , respectively; the rotating frames o 0 1 x @ 1 y @ 1 and o 0 2 x @ 2 y @ 2 , that dedicates the rotation motion around points o 0 1 and o 0 2 , respectively. The six reference frames of the vibro-bodies separately coincide with each other when the system is in the static equilibrium state. Since the two vibro-bodies are supported by two elastic foundations, it exhibits six degrees of freedom. The mass center coordinates of vibro-bodies x 1 , x 2 , y 1 and y 2 , as well as the rotation coordinates ψ 1 and ψ 2 , are set as the independent coordinates. The unbalanced rotors rotate about their own spin axes, which are denoted by φ 1 and φ 2 , respectively. In reference frame o ̋ x ̋ y ̋ , the coordinates of each exciter, Φ @ , can be described ...

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... shown in Fig 1, the phase angular of the rotors are defined as follows: ...