scholarly journals Synchronization of Two Non-Identical Coupled Exciters in a Non-Resonant Vibrating System of Linear Motion. Part I: Theoretical Analysis

2009 ◽  
Vol 16 (5) ◽  
pp. 505-515 ◽  
Author(s):  
Chunyu Zhao ◽  
Hongtao Zhu ◽  
Ruizi Wang ◽  
Bangchun Wen
Keyword(s):  
Zero Solution ◽  
Moment Of Inertia ◽  
Moments Of Inertia ◽  
Vibrating System ◽  
Frequency Capture ◽  
Synchronization Problem ◽  
Small Parameters ◽  
Existence And Stability ◽  
Synchronous Operation ◽  
The Stability

In this paper an analytical approach is proposed to study the feature of frequency capture of two non-identical coupled exciters in a non-resonant vibrating system. The electromagnetic torque of an induction motor in the quasi-steady-state operation is derived. With the introduction of two perturbation small parameters to average angular velocity of two exciters and their phase difference, we deduce the Equation of Frequency Capture by averaging two motion equations of two exciters over their average period. It converts the synchronization problem of two exciters into that of existence and stability of zero solution for the Equation of Frequency Capture. The conditions of implementing frequency capture and that of stabilizing synchronous operation of two motors have been derived. The concept of torque of frequency capture is proposed to physically explain the peculiarity of self-synchronization of the two exciters. An interesting conclusion is reached that the moments of inertia of the two exciters in the Equation of Frequency Capture reduce and there is a coupling moment of inertia between the two exciters. The reduction of moments of inertia and the coupling moment of inertia have an effect on the stability of synchronous operation.

scholarly journals Synchronization of Two Non-Identical Coupled Exciters in a Non-Resonant Vibrating System of Linear Motion. Part II: Numeric Analysis

2009 ◽  
Vol 16 (5) ◽  
pp. 517-528 ◽  
Author(s):  
Chunyu Zhao ◽  
Hongtao Zhu ◽  
Tianju Bai ◽  
Bangchun Wen
Keyword(s):  
Phase Difference ◽  
Structural Parameters ◽  
Load Torque ◽  
Equivalent Radius ◽  
Vibrating System ◽  
Frequency Capture ◽  
Coupling Dynamic ◽  
Synchronous Operation ◽  
The Stability ◽  
Phase Angles

The paper focuses on the quantitative analysis of the coupling dynamic characteristics of two non-identical exciters in a non-resonant vibrating system. The load torque of each motor consists of three items, including the torque of sine effect of phase angles, that of coupling sine effect and that of coupling cosine effect. The torque of frequency capture results from the torque of coupling cosine effect, which is equal to the product of the coupling kinetic energy, the coefficient of coupling cosine effect, and the sine of phase difference of two exciters. The motions of the system excited by two exciters in the same direction make phase difference close to π and that in opposite directions makes phase difference close to 0. Numerical results show that synchronous operation is stable when the dimensionless relative moments of inertia of two exciters are greater than zero and four times of their product is greater than the square of their coefficient of coupling cosine effect. The stability of the synchronous operation is only dependent on the structural parameters of the system, such as the mass ratios of two exciters to the vibrating system, and the ratio of the distance between an exciter and the centroid of the system to the equivalent radius of the system about its centroid.

scholarly journals Theoretical and Experimental Study on Synchronization of the Two Homodromy Exciters in a Non-Resonant Vibrating System

2013 ◽  
Vol 20 (2) ◽  
pp. 327-340 ◽  
Author(s):  
Xue-Liang Zhang ◽  
Chun-Yu Zhao ◽  
Bang-Chun Wen
Keyword(s):  
Phase Difference ◽  
Structural Parameters ◽  
Circular Motion ◽  
Vibrating System ◽  
Frequency Capture ◽  
Rigid Frame ◽  
Small Parameters ◽  
Coupling Dynamic ◽  
The Stability ◽  
Elliptical Motion

In this paper we give some theoretical analyses and experimental results on synchronization of the two non-identical exciters. Using the average method of modified small parameters, the dimensionless coupling equation of the two exciters is deduced. The synchronization criterion for the two exciters is derived as the torque of frequency capture being equal to or greater than the absolute value of difference between the residual electromagnetic torques of the two motors. The stability criterion of synchronous state is verified to satisfy the Routh-Hurwitz criterion. The regions of implementing synchronization and that of stability of phase difference for the two exciters are manifested by numeric method. Synchronization of the two exciters stems from the coupling dynamic characteristic of the vibrating system having selecting motion, especially, under the condition that the parameters of system are complete symmetry, the torque of frequency capture stemming from the circular motion of the rigid frame drives the phase difference to approach PI and carry out the swing of the rigid frame; that from the swing of the rigid frame forces the phase difference to near zero and achieve the circular motion of the rigid frame. In the steady state, the motion of rigid frame will be one of three types: pure swing, pure circular motion, swing and circular motion coexistence. The numeric and experiment results derived thereof show that the two exciters can operate synchronously as long as the structural parameters of system satisfy the criterion of stability in the regions of frequency capture. In engineering, the distance between the centroid of the rigid frame and the rotational centre of exciter should be as far as possible. Only in this way, can the elliptical motion of system required in engineering be realized.

Thermal effects on the figure of the Moon

1967 ◽  
Vol 296 (1446) ◽  
pp. 266-269 ◽  
Keyword(s):  
Thermal Effects ◽  
Hydrostatic Equilibrium ◽  
Moment Of Inertia ◽  
The Moon ◽  
Moments Of Inertia ◽  
The Earth ◽  
Principal Moments Of Inertia ◽  
Long Time ◽  
The Stability ◽  
Lunar Rotation

Before discussing its cause, one must be clear in exactly what respect the lunar figure deviates from the equilibrium one. This is necessary because there has been confusion over the question for a long time. It was known early that the Moon’s ellipsoid of inertia is triaxial and that the differences of the principal moments of inertia determined from observations are several times larger than the theoretical values corresponding to hydrostatic equilibrium. The stability of lunar rotation requires that the axis of least moment of inertia point approximately towards the Earth and the laws of Cassini show that it is really so.

scholarly journals Decomposition and stability of linear singularly perturbed systems with two small parameters

2021 ◽  
Vol 13 (1) ◽  
pp. 15-21
Author(s):  
O.V. Osypova ◽  
A.S. Pertsov ◽  
I.M. Cherevko
Keyword(s):  
Original System ◽  
Zero Solution ◽  
Singularly Perturbed ◽  
Reduction Principle ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Perturbed System ◽  
Integral Manifolds ◽  
Small Parameters ◽  
The Stability

In the domain $\Omega =\left\{\left(t,\varepsilon _{1}, \varepsilon _{2} \right): t\in {\mathbb R},\varepsilon _{1}>0, \varepsilon _{2} >0\right\}$, we consider a linear singularly perturbed system with two small parameters \[ \left\{ \begin{array}{l} {\dot{x}_{0} =A_{00} x_{0} +A_{01} x_{1} +A_{02} x_{2},} \\ {\varepsilon _{1} \dot{x}_{1} =A_{10} x_{0} +A_{11} x_{1} +A_{12} x_{2},} \\ {\varepsilon _{1} \varepsilon _{2} \dot{x}_{2} =A_{20} x_{0} +A_{21} x_{1} +A_{22} x_{2},} \end{array}\right. \] where $x_{0} \in {\mathbb R}^{n_{0}}$, $x_{1} \in {\mathbb R}^{n_{1}}$, $x_{2} \in {\mathbb R}^{n_{2}}$. In this paper, schemes of decomposition and splitting of the system into independent subsystems by using the integral manifolds method of fast and slow variables are investigated. We give the conditions under which the reduction principle is truthful to study the stability of zero solution of the original system.

scholarly journals Synchronization of Two Self-Synchronous Vibrating Machines on an Isolation Frame

2011 ◽  
Vol 18 (1-2) ◽  
pp. 73-90 ◽  
Author(s):  
Chunyu Zhao ◽  
Qinghua Zhao ◽  
Zhaomin Gong ◽  
Bangchun Wen
Keyword(s):  
Electromagnetic Torque ◽  
Frequency Capture ◽  
Generalized System ◽  
Rigid Frame ◽  
Generalized Systems ◽  
Three Phase ◽  
Small Parameters ◽  
Angular Velocities ◽  
The Absolute ◽  
The Stability

This paper investigates synchronization of two self-synchronous vibrating machines on an isolation rigid frame. Using the modified average method of small parameters, we deduce the non-dimensional coupling differential equations of the disturbance parameters for the angular velocities of the four unbalanced rotors. Then the stability problem of synchronization for the four unbalanced rotors is converted into the stability problems of two generalized systems. One is the generalized system of the angular velocity disturbance parameters for the four unbalanced rotors, and the other is the generalized system of three phase disturbance parameters. The condition of implementing synchronization is that the torque of frequency capture between each pair of the unbalanced rotors on a vibrating machine is greater than the absolute values of the output electromagnetic torque difference between each pair of motors, and that the torque of frequency capture between the two vibrating machines is greater than the absolute value of the output electromagnetic torque difference between the two pairs of motors on the two vibrating machines. The stability condition of synchronization of the two vibrating machines is that the inertia coupling matrix is definite positive, and that all the eigenvalues for the generalized system of three phase disturbance parameters have negative real parts. Computer simulations are carried out to verify the results of the theoretical investigation.

Vibratory synchronization transmission of two cylindrical rollers in a super-resonant vibrating system with symmetrical structure

2018 ◽  
Vol 233 (4) ◽  
pp. 1204-1223
Author(s):  
Dawei Gu ◽  
Bangchun Wen ◽  
Juqian Zhang
Keyword(s):  
Dry Friction ◽  
Theoretical Basis ◽  
Averaging Method ◽  
Project Design ◽  
System Parameters ◽  
Vibrating System ◽  
Vibratory Synchronization Transmission ◽  
Small Parameters ◽  
The Stability ◽  
Good Agreement

In this paper, vibratory synchronization transmission of two cylindrical rollers (CRs) with dry friction in a vibrating system excited by two co-rotating exciters is investigated. By virtue of the averaging method of modified small parameters, the synchronization criterion for two exciters and the criterion of vibratory synchronization transmission for two cylindrical rollers are obtained. The stability criterion of synchronous state is acquired based on the Routh–Hurwitz criterion. The effects of system parameters on the ability of synchronization, vibratory synchronization transmission and stability, are quantitatively discussed, which provide a theoretical basis for the project design. Finally, an experiment on a corresponding vibratory synchronization transmission bench is performed to examine the results of the theoretical analysis and numerical discussion. Experimental results will show good agreement with the numerical results. In engineering, certain types of vibratory crusher or vibratory mill can be developed according to the vibratory synchronization transmission theory of two cylindrical rollers.

scholarly journals Selected synchronous state of the vibration system driven by three homodromy eccentric rotors

2019 ◽  
Vol 39 (2) ◽  
pp. 352-367
Author(s):  
Xiaozhe Chen ◽  
Lingxuan Li
Keyword(s):  
External Disturbance ◽  
Plane Motion ◽  
Initial Condition ◽  
Vibration System ◽  
Synchronous State ◽  
Frequency Capture ◽  
Synchronous Motion ◽  
Small Parameters ◽  
Coupling Dynamic ◽  
The Stability

In order to verify that the vibration system driven by multi-eccentric rotors has multiple synchronous states, a model of three eccentric rotors horizontal installation of plane motion is established to study the coupling dynamic characteristics. Based on the average method of modified small parameters, the frequency capture equation is constructed to obtain the conditions of synchronous motion and vibration synchronization transmission. According to the synchronous conditions, the stability state of achieving synchronous motion is converted into the solution of balance equation of the synchronous torque. There are multiple solutions to the torque equation because of the non-linear characteristics of the vibration system driven by multi-eccentric rotors. Then the stability condition is used to estimate which of the solutions is stable. After substituting the parameters of the experimental machine into the above method, the curves of the phase difference and its stability coefficients of three eccentric rotors system are obtained numerically. Two experiments show that the selected synchronous state depends on the initial condition and external disturbance, and if the vibration synchronization transmission conditions are satisfied, three eccentric rotors can not only achieve vibration synchronization transmission of one motor with switching off the power but also that of two motors with switching off the power.

scholarly journals Synchronization of Two Asymmetric Exciters in a Vibrating System

2011 ◽  
Vol 18 (1-2) ◽  
pp. 63-72 ◽  
Author(s):  
Zhaohui Ren ◽  
Qinghua Zhao ◽  
Chunyu Zhao ◽  
Bangchun Wen
Keyword(s):  
Average Method ◽  
Structural Parameters ◽  
Positive Definite ◽  
Electromagnetic Torque ◽  
Coupling Matrix ◽  
Vibrating System ◽  
Frequency Capture ◽  
Small Parameters ◽  
The Difference ◽  
Dimensional Coupling

We investigate synchronization of two asymmetric exciters in a vibrating system. Using the modified average method of small parameters, we deduce the non-dimensional coupling differential equations of the two exciters (NDDETE). By using the condition of existence for the zero solutions of the NDDETE, the condition of implementing synchronization is deduced: the torque of frequency capture is equal to or greater than the difference in the output electromagnetic torque between the two motors. Using the Routh-Hurwitz criterion, we deduce the condition of stability of synchronization that the inertia coupling matrix of the two exciters is positive definite. A numeric result shows that the structural parameters can meet the need of synchronization stability.

Research on the Simulation of Self-Synchronization of Dual Mass for Vibrating System with Two-Motor Drives

2013 ◽  
Vol 300-301 ◽  
pp. 18-21 ◽  
Author(s):  
Duo Yang ◽  
Ye Li ◽  
He Li ◽  
Bang Chun Wen
Keyword(s):  
Dynamic Characteristics ◽  
Structural Parameters ◽  
System Simulation ◽  
Motor Drives ◽  
The Self ◽  
Vibrating System ◽  
Coupling Dynamic ◽  
Synchronous Operation ◽  
The Difference ◽  
The Stability

The coupling dynamic characteristics of the vibrating system with dual mass are analyzed quantitatively. Through numerical computation, the effects of translation and rotation in the system regarding self-synchronization are discussed. The phase difference of two eccentric blocks is caused by the difference of the rated revolution of two motors. The stability of the synchronous operation is dependent on the structural parameters of the system. Simulation is carried out to verify that the system can be synchronized and the model can guarantee the stability of synchronization if the parameters of the system meet the conditions of synchronous implementation and stability. Simulations are also performed for the self-synchronization of two motors with different rated revolutions.

Optimization Design of a Vibrating System with Two Motors

2009 ◽  
Vol 628-629 ◽  
pp. 67-72
Author(s):  
D.G. Wang ◽  
K. Guo ◽  
Chun Yu Zhao ◽  
Bang Chun Wen
Keyword(s):  
Dynamic Analysis ◽  
Dynamic Model ◽  
Computer Simulations ◽  
Optimization Design ◽  
Phase Synchronization ◽  
Simulation Program ◽  
Synchronous State ◽  
Vibrating System ◽  
Frequency Capture ◽  
Synchronous Operation

The dynamic model of vibrating system with two motors is established. Through dynamic analysis, the equations of frequency capture of the vibrating system and the conditions of implementing stable self-synchronous operation are obtained. Then the vibrating system is optimization designed based on the conditions of implementing stable self-synchronous operation. The simulation program with proper parameters of vibrating system is run, and the results show that the system is in a good synchronous state. Computer simulations demonstrate that the vibrating system realizes speed synchronization and phase synchronization. The results verify the effectiveness of the optimization design.

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