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 vibrobody and the pendulum rod is ascertained with the Laplace transformation method, based on the dimensionless equation of the dynamics system. The Poincare method is employed to study the synchronization state between the two unbalanced rotors, which is converted into that of existence and the stability of solutions for synchronization-balance equations. The obtained results are supported by computer simulations. It is demonstrated that the values of the spring stiffness coefficient, the length of the pendulum, and the angular installation of the pendulum are important parameters with respect to the synchronous behavior in the rotor-pendulum system.
The word “synchronization” is often encountered in both science and daily life. Our surroundings are full of synchronization phenomenon, which is considered as an adjustment of rhythms of oscillating objects due to their internal weak couplings [
The abovementioned researches are mainly synchronization of the pendula or the rotors; however, the synchronization of the rotors coupled with pendula is less reported. Recently, we have proposed synchronization of two homodromy rotors coupled with a pendulum rod in an postresonant system [
This paper is organized as follows. Section
Consider the dynamic equation of a rotation system:
Based on ( Steady forced vibrations with from the supporting body or supporting system of bodies (i.e., from the second formula of ( Equation ( Here and below the angle brackets where symbol If a certain set of constants are negative, then at sufficiently small
A simplified rotor-pendulum system depicted in Figure
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.
As illustrated in Figure
In the reference frame
In the reference frame
The kinetic energy of the system can be obtained by
The potential energy of the system is approximated to
Moreover, the viscous dissipation function
The dynamics equation is obtained by using Lagrange’s equation:
If
Assuming that the natural frequency of the system is unequal to or far away from the excitation frequency, thus the oscillating angle
Having calculated the derivative of the function
The first four formulas in (
Applying the inverse Laplace transformation to (
Parameter
Parameter values.
Parameter values for system equation (
Unbalanced rotor | Vibrobody | Pendulum rod | Induction motor |
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Parameter values according to dimensionless equation (
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From Figure
Type of the coupling system.
Abbreviation | SBCWB | SACWB | SBCWA | SACWA |
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Full name | System of before-resonance coupled with before-resonance | System of after-resonance coupled with before-resonance | System of before-resonance coupled with after-resonance | System of after-resonance coupled with after-resonance |
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Frequency ratio |
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The value of the coupling coefficients of the system for
In this section, we analyze the synchronization and stability of the system with theoretical method. As was already mentioned in (
Now, let us consider the stability of the synchronous rotation for the rotors. Considering (
The abovementioned sections have given some theoretical discussions in the simplified form on synchronization problem for the vibration system that the two unbalanced rotors coupled with a pendulum rod. In this section, we will quantitatively analyze the results of the stable phase difference. The parameter values corresponding to general engineering application are as given in Table
Under the condition that the balance equation and stability criterion of synchronization between the two rotors (see (
Depending on variations of the values of the system parameters, in this subsection, we observe the synchronous states of two different systems, that is, system of before-resonance coupled with before-resonance (SBCWB), and system of after-resonance coupled with before-resonance (SACWB). We choose the systems as the comparison of the synchronous states because we want to know whether the variations of the stiffness coefficients of the springs have influence on the synchronous states, although SBCWB may be rarely applied in vibration screening engineering.
In Figure
Stable phase difference for SBCWB and SACWB considering variation of frequency ratio
For synchronization of asymmetric mass of the two rotors (i.e., considering
Stable phase difference for SBCWB and SACWB considering asymmetric mass of the rotors with
On the other hand, whether length
Stable phase difference for SACWB considering the variation of length
In the above subsection, we have computed the stable phase difference for SBCWB and SACWB considering variations of the structure parameters. Here, we will concern the phase difference for the system of before-resonance coupled with after-resonance (SBCWA, i.e.,
In Figure
Stable phase difference for SBCWA and SACWA considering
From Figure
Stable phase difference for SBCWA and SACWA assuming the nonidentical mass of rotors and considering
Finally, the variations of the phase difference for SACWA are compared in Figure
Stable phase difference for SACWA assuming the nonidentical mass of rotors and considering
Further analyses have been performed by computer simulations to verify our theoretical solutions above, which can be carried out by applying the Runge-Kutta routine with adaptive stepsize control to the dynamics equation (
Simulation results for SACWB, assuming that
Simulation results for SACWB: (a) rotational velocity, (b) phase difference of the two rotors, (c) torques of the two rotors, (d) trajectory of mass center of the vibrobody, (e) displacement responses of the vibrobody in
Simulation results for SACWA, assuming that the values of system parameters are in accordance with Section
Simulation results for SACWA: (a) rotational velocity, (b) phase difference of the two rotors, (c) torques of the two rotors, (d) trajectory of mass center of the vibrobody, (e) displacement responses of the vibrobody in
To verify the synchronization problem of the asymmetry rotors, we take into account that the mass of the rotor, connected with pendulum rod, is nonidentical to another rotor. This simulation is performed on the assumption that
Simulation results for asymmetry rotors system: (a) rotational velocity, (b) phase difference of the two rotors, (c) torques of the two rotors, (d) trajectory of mass center of the vibrobody, (e) displacement responses of the vibrobody in
The occurrence of synchronized motion in the rotor-pendulum system via the multi-DOF vibration has been investigated. Firstly, the dynamics equation of the system, deduced by Lagrange equation, has been converted into the dimensionless equations. Then Laplace transformation has been employed to solve the steady responses of the system. Next, the balance equation and the stability criteria of synchronization of the system have been derived using Poincare method base on the small parameter. Only should the values of the structure parameters satisfy the balance equation and the stability criteria, the stable phase difference can be acquired. Finally, the value of the phase difference has been calculated with numerical method; moreover, computer simulations have been performed to verify the correctness of the theoretical computations.
The analysis has revealed that the synchronous state of the rotor-pendulum system is mainly sensitive to the values of the spring stiffness coefficient, the length, and the angular installation of the pendulum rod. However, the synchronous state of the system is independent of the mass ratio between the two rotors and vibrobody, besides the operation velocity of the motors. With regard to the displacement amplitude of the vibrobody, the amplitude of the vibrobody in synphase synchronization is larger than antiphase synchronization. Meanwhile, reasonable values of the structure parameter are in favour of the implementing elliptical motion with large amplitude for vibrobody. Therefore, the vibration system proposed in this paper can be applied to design new balanced elliptical vibrating screens, when their structure parameters satisfy the balance equation and the stability criterion of synchronization. In the early stage, for the developing and understanding the internal characteristics of the system, we only consider the rotors under the assumption of the rotation in the same direction. Will these results change if the contra rotation of the rotors in the pendulum-rotor system is taken into account? We believe that finding the answer to this question is the next step in challenging task of getting a complete understanding of synchronization in such system.
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) and Key Project of Talent Engineering of Sichuan (Grant no. 2016RZ0059).