On the synchronization of chains of nonlinear pendula connected by linear springs

Proceedings of the Institution of Mechanical Engineers, Part C:

2017

The problem is motivated by observations of a rotor-pendula system, which derived from a new shale shaker. To grasp the dynamic characteristics of the shale shaker, the key research is exploring the synchronous mechanism for the system, since synchronous state between rotors is closely related to the dynamic characteristics of the system. In this paper, the dynamic equation of the rotor-pendula system is firstly derived by applying Lagrange’s equations. Through Laplace’s transformation method, the approximate responses of the system in synchronous state are obtained, which is determined coupling coefficients and synchronous state of the system. Then, the synchronous balance equation and the stability criterion of the system are obtained with Poincaré method on which stable phase difference and synchronous behavior can be ascertained. To verify the correctness of the theoretical analysis, numerical simulations are implemented by Runge–Kutta method, and it is shown that the synchronous behavior is determined by the geometry parameters, coupling coefficients, and rotor rotation direction.

“…Substituting equations (10), (11), (12), and 14into equation 14, considering m i ( m 0 , ( 1, ' 1 ( 1, the dynamic equation of the system is given in the following:…”

Section: Simplified Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Synchronization characteristics of a rotor-pendula system in multiple coupling resonant systems

Proceedings of the Institution of Mechanical Engineers, Part C:

2017

“…It seems that the first observation and description of a specific case of synchronization—the mutual synchronization of pendula clocks—was made by Christian Huygens 8 in 1673. Nowadays, the synchronization of derived models of Huygens’ clocks, such as the coupled pendula, 9 pendula coupled by elastic elements, etc., 10,11 are also the problems of important concern for many scholars, and the authors illustrated that the pendula could appear simultaneously in rotation and in synchronous vibration. In 1950, another typical case of synchronization, two unbalanced rotors mounted on the same vibrating organ driven by asynchronization motors, is proposed by Signeul, 12 and it was found that synchronization was occurring despite possible parameter differences and the absence of any kinematic or electrical connections between the rotors.…”

Section: Introductionmentioning

confidence: 99%

Synchronization behavior of triple-rotor-pendula system in a dual-super-far resonance system

Proceedings of the Institution of Mechanical Engineers, Part C:

2018

In this paper, a novel design for vibrating screens, called the triple-rotor-pendula system, is presented in a dual-super-far resonance system, which makes it possible to produce wanted vibrations with self-adjustable of the synchronization state between unbalanced rotors. To grasp the synchronization characteristics of the system, the dynamics equation of the triple-rotor-pendula system is primarily derived by adopting Lagrange’s equations. Next, the displacement responses of the system in steady state are obtained by Laplace’s transform method. Meanwhile, considering the average method with revised small parameter, the synchronization implement of the system is ascertained on condition that the absolute values of residual electromagnetism torques between two arbitrary motors are less than or equal to their load torque differences. Then, according to Routh–Hurwitz theorem and generalized Lyapunov equation, the criterion of synchronization stability of the system is obtained. Finally, the simulation computations confirm the results of analytical investigations, which show that the theoretical analysis for the synchronization behavior of the triple-rotor-pendula system is feasible. It can be known that the synchronization state of the system is influenced by mass ratio coefficients, structure parameters, rotating directions, and frequency ratios.

“…The so-called synchronization phenomenon refers to self-adjusting different frequencies of oscillating objects to a unified frequency relying on their internal weak couplings [1,2]. In recent years, the study of synchronization phenomenon is involved in the fields of physics, chemistry, and biology, such as the practical application on complex dynamic network systems [3][4][5], nonlinear coupled chaotic systems [6,7], coupled pendulum system [8][9][10][11][12], and rotor system [1,[13][14][15][16][17][18][19]. Nevertheless, the latter two systems can be categorized as synchronous problem of mechanics system in detail.…”

Section: Introductionmentioning

confidence: 99%

“…But mass of the motor and the rotor, which can be relatively heavy or light, may affect the system motion. Therefore, the proposed problem is different from the dynamics of classical pendulum systems presented in papers [8][9][10][11] and classical rotor systems presented in papers [12][13][14][15][16][17][18][19][20][23][24][25][26][27]. We consider an ideal source of energy supply to asynchronous motors, which drag the unbalanced rotors.…”

Section: Introductionmentioning

confidence: 99%

Theoretical Study of Synchronous Behavior in a Dual‐Pendulum‐Rotor System

Dai

et al. 2018

Shock and Vibration

A dual-pendulum-rotor system widely appears in aero-power plant, mining screening machines, parallel robots, and the like of the other rotation equipment. Unfortunately, the synchronous behavior related to the dual-pendulum-rotor system is less reported. Based on the special backgrounds, a simplified mechanical model of the dual-pendulum-rotor system is proposed in the paper, and the intrinsic mechanisms of synchronous phenomenon in the system are further revealed with employing the Poincaré method. The research results show that the spring stiffness, the installation angular of the motor, and rotation direction of the rotors have a large influence on the existence and stability of the synchronization state in the coupling system, and the mass ratios of the system are irrelevant to the synchronous state of the system. It should be noted that to ensure the implementation of the synchronization of the system, the values of the parameters of the system should be far away to the two "critical points".