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The model of the vibration system: (a) dynamic model of the double vibrobody system with two induction motors rotating in the same direction and (b) the reference frame system.
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This work is a continuation for our published literature for vibration synchronization. A new mechanism, two rotors coupled with a pendulum rod in a multi-DOF vibration system, is proposed to implement coupling synchronization, and the dynamics equation of mechanism is derived by Lagrange equation. In addition, the coupling relationship between the...
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From the perspective of theoretical derivation, numerical simulation, and engineering application, the vibratory synchronization characteristics of a dual-mass vibrating system driven by two exciters, were studied. The differential motion equations of the total system were calculated using Lagrange’s equations, and the responses of the vibrating sy...
Citations
... The research on rotating pendula involves various types of models and arrangements, just to mention rotating hubs [4], double [5,6] and spherical [7] pendula or multi-DOF systems [8]. The studies on slowly rotating nodes can be found in [9,10], while in [11] Authors investigate the dual-pendulum-rotor scheme. ...
In this paper, we investigate the complex dynamics of rotating pendula arranged into a simple mechanical scheme. Three nodes forming the small network are coupled via the horizontally oscillating beam (the global coupling structure) and the springs (the local coupling), which extends the research performed previously for similar models. The pendula rotate in different directions, and depending on the distribution of the latter ones, various types of behaviors of the system can be observed. We determine the regions of the existence and co-existence of particular solutions using both the classical method of bifurcations, as well as a modern sample-based approach based on the concept of basin stability. Various types of states are presented and discussed, including synchronization patterns, coherent dynamics, and irregular motion. We uncover new schemes of solutions, showing that both rotations and oscillations can co-exist for various pendula, arranged within one common system. Our analysis includes the investigations of the basins of attraction of different dynamical patterns, as well as the study on the properties of the observed states, along with the examination of the influence of system's parameters on their behavior. We show that the model can respond in spontaneous ways and uncover unpredicted irregularities occurring for the states. Our study exhibits that the inclusion of the local coupling structure can induce complex, chimeric dynamics of the system, leading to new co-existing patterns for coupled mechanical nodes.
... Fang et al. proposed a Rotor-Pendulum System with the Multi-DOF Vibration, and the Poincare method was employed to study the synchronization characteristics of the system in a far-resonant vibrating system. It is indicated that the stiffness of the support spring, the stiffness of the connecting spring and the installation location of the motors affect the synchronous state of the system [25][26][27]. However, for synchronization of two exciters rotating with the same direction, the relationships between the phase difference of two exciters ( ) 2 and compound exciting force on the screen ( ) F are shown in Figure 1. ...
This work is a continuation and verification of the original literature by using experimental strategy. Based on the published paper, in order to avoid anti-phase synchronization with two co-rotating rotors system, a vibrating system with two co-rotating rotors installed with nonlinear springs have been proposed, and then, the synchronous condition and the synchronous criterion of the system are theoretically derived. From the analysis mentioned, it is shown that the synchronous state is mainly determined by the structural parameters of the coupling unit, coupling coefficients and positional parameters of the two exciters, etc. The main objective of the present work is to investigate the synchronous mechanism by experiments and simulations in this paper. Some simulation computations are firstly implemented to explain the synchronous mechanism of the system. Additionally, an experimental strategy with synchronous tests and dynamic characteristic tests of the vibrating system are carried out to validate the correctness of the simulation analysis. The simulations and experiments demonstrate that the nonlinear springs can overcome the difference of residual torques of the two motors to realize the synchronization of near zero phase difference under the condition of in-phase difference between two exciters. Finally, the error analysis results among the dynamic testing, synchronous testing results and simulations are discussed. This research can provide theoretical reference for designing large-sized and heavy-duty Vibrating Screens.
... During the recent years, the self-synchronization of the coupled pendulum or mechanical rotors has attracted many researchers' attention. Fang employed Poincare method to explore the synchronization for two rotors coupled with pendulum rod through a torsion spring in a far-resonant vibrating system [17,18]. Kumon et al. investigated the properties of nonlinear oscillations of two 1-DOF pendulums coupled with a weak spring controlled by utilizing SGE controller [19]. ...
In this paper, the synchronization of a vibrating mechanism driven by two co-rotating motors coupled with a tensile-spring in a non-resonant system is investigated. In order to master the synchronous behavior of the vibrating system, the dynamic model of the two motors coupled with a tensile-spring is firstly established, and the differential equations of the dynamic system are derived by using Lagrange's equations. Second, a typical approach, called as method of direct separation of motions, is employed to explore the synchronizing characteristic of the vibrating system. Accordingly, the differential equations of motion are transformed into two components: the "fast" component and the "slow" component. Subsequently, the synchronous condition and the synchronous stability of the system are deduced by the first-order linear approximation. In addition, the key factors of the rigidity coefficient of the springs and positional parameters of two motors are taken into account, and a numerical analysis on the influence of the stable phase difference in synchronous state is presented. Finally, computer simulations are proved to be in good agreement with the theoretical analysis. It is demonstrated that the tensile-spring can make the phase difference between the two rotors close to zero by the selection of a large enough stiffness.
... 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. ...
... Based on the approximate solutions above, the synchronization and stability of the system can be determined with Poincaré method [20,[25][26][27]. In addition, the phase angle of the rotors can be assumed by ...
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”.
... In [13][14][15][16][17][18][19], motion stability, corresponding to the pendulums or the balls getting stuck, is investigated analytically by Lyapunov (in a small) in combination with methods of the small parameter: in [13][14][15][16][17] -using the method of synchronization of dynamical systems [13]; in [18,19] -applying the method of separation of motions [10]. ...
... In [13][14][15][16][17][18][19], motion stability, corresponding to the pendulums or the balls getting stuck, is investigated analytically by Lyapunov (in a small) in combination with methods of the small parameter: in [13][14][15][16][17] -using the method of synchronization of dynamical systems [13]; in [18,19] -applying the method of separation of motions [10]. ...
... Authors of [13][14][15] examined the effect of pendulums getting stuck in vibratory machines. For a pendulum, mounted on the electric motor shaft, placed on the platform of a vibratory machine, studies were conducted for the following cases: a low-power electric motor [13]; an electric motor whose rated rotation frequency slightly exceeds resonance frequency of oscillations of the platform [14]. ...
By employing computational experiments, we investigated stability of the dual-frequency modes of motion of a single-mass vibratory machine with translational rectilinear motion of the platform and a vibration exciter in the form of a passive auto-balancer.
For the vibratory machines that are actually applied, the forces of external and internal resistance are small, with the mass of loads much less than the mass of the platform. Under these conditions, there are three characteristic rotor speeds. In this case, at the rotor speeds:
– lower than the first characteristic speed, there is only one possible frequency at which loads get stuck; it is a pre-resonance frequency;
– positioned between the first and second characteristic speeds, there are three possible frequencies at which loads get stuck, among which only one is a pre-resonant frequency;
– positioned between the second and third characteristic speeds, there are three possible frequencies at which loads get stuck; all of them are the over-resonant frequencies;
– exceeding the third characteristic speed, there is only one possible frequency at which loads get stuck; it is the over-resonant frequency and it is close to the rotor speed.
Under a stable dual-frequency motion mode, the loads: create the greatest imbalance; rotate synchronously as a whole, at a pre-resonant frequency. The auto-balancer excites almost perfect dual-frequency vibrations. Deviations of the precise solution (derived by integration) from the approximated solution (established previously using the method of the small parameter) are equivalent to the ratio of the mass of loads to the mass of the entire machine. That is why, for actual machines, deviations do not exceed 2 %.
There is the critical speed above which a dual-frequency motion mode loses stability. This speed is less than the second characteristic speed and greatly depends on all dimensionless parameters of the system.
At a decrease in the ratio of the mass of balls to the mass of the entire system, critical speed tends to the second characteristic speed. However, this characteristic speed cannot be used for the approximate computation of critical speed due to an error, rapidly increasing at an increase in the ratio of the mass of balls to the mass of the system. Based on the results of a computational experiment, we have derived a function of dimensionless parameters, which makes it possible to approximately calculate the critical speed.
This paper addresses unbalanced trajectories of dual-motor vibrating screens by proposing a dynamic theory and experimental testing. Firstly, based on the Lagrange equation, the dynamic model of ipsilateral offset dual-motor excitation system is established. Subsequently, the steady response is worked out by solving the differential equation of motion of the vibrating body. Then, the synchronization condition and synchronization stability criterion of the system are determined employing the small parameter method and the Routh-Hurwitz criterion. Numerical analysis methods are given to reveal the relationship between structural parameters and synchronization characteristics of the vibration system. Moreover, an mechanical-electric coupling model is developed to study dynamic characteristics. The results demonstrate that synchronization condition can be described with capture torque and difference of output torque between motors; The synchronization ability is enhanced by improving installation distance of the motors; Additionally, Increasing angle β or decreasing angle θ of the motors is in favor of optimization of centroid trajectory; swing oscillation of the box can be controlled by adjusting the main inclination angle of motors; in such a condition, equilibrium of motion trajectory of system is enhanced. Finally, the validity method for designing vibrating screen is verified through theoretical calculations, simulation analysis, and experiments.
This paper researched the synchronous characteristic of three homodromy motors in a secondary isolation system. Firstly, the motion differential equations of the system were established via employing Lagrange’s equation. The Laplace transform was used to deduce approximate solutions of displacement. Next, the ability of vibrating isolation was determined through amplitude ratio. Secondly, the synchronous condition and steady criterion was obtained via employing the average small parameters method and Lyapunov criterion. And then the influence of structural parameters of the system on synchronous ability was analyzed. Moreover, the synchronous characteristics of the system, including the difference of residual torque among induction motors, the synchronous ability, the output torque of motors and the stable phase difference, were discussed through numerical analyses. Finally, the accuracy of theoretical analyses was proved through computer simulation results. It is shown that, the ability of vibrating isolation is principally impacted by vertical spring stiffness; the greater the installation distance among three motors, the easier to achieve synchronous motion of the rotors; meanwhile, the output torque of asynchronous motors is affected by the synchronous state of the vibrating system; the synchronous characteristic is mainly affected by the mounting dip angle of motors and the installation distance among motors. The research results can promote development of vibration engineering and related fields.
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 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.
