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Study on stability of self-synchronous far-resonant vibrating system of two eccentric rotors considering material impact

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Abstract

The self-synchronizing far-resonant vibrating system of two eccentric rotors is widely used in petroleum, mining and food industries, and its motion stability is affected by material impact. However, the synchronous characteristics and stability of this kind of the system are studied rarely. Based on the background, a simplified mechanical model of the self-synchronous vibrating system driven by two eccentric rotors considering material impact is proposed. Firstly, the differential equations of motion about the system in non-collision and collision phase are established by using Lagrange equation and the theorem of momentum. Then, the section of Poincare maps and linearization matrix at the fixed point are solved. Finally, the dynamic behavior of material and system is analyzed, and then the change characteristics of the phase difference of the two eccentric rotors are revealed by numerical simulation. It can be concluded that the motion forms of the system and the rules of abrupt change about the phase difference evolve from periodic variation into chaotic state with the mass ratio of material to vibrating body increasing.

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Abbreviations

m 0 :

Mass of vibrating body

m i :

Mass of eccentric rotor i (i = 1, 2)

m 3 :

Mass of material

M :

The total mass of the vibrating system \(M = \sum\nolimits_{i = 0}^2 {{m_i}} \).

g :

Acceleration of gravity

k x :

Stiffness coefficient of spring

c :

Damping coefficient of damper

r i :

Rotation radius of eccentric rotor i (i = 1, 2)

r :

Rotation radius of eccentric rotors

R :

Recovery coefficient

β :

Installation angle

φ i :

Angular displacement of eccentric rotor i (i = 1, 2)

φ :

The average phase angle

ω m :

Average speed of the eccentric rotor in a cycle

ρ x :

The amplitude amplification factor

Y x :

The phase lag angle

α 1 :

Phase difference of the two eccentric rotors

α 1 :

Initial phase difference of the two eccentric rotors

ω x :

Natural frequency of the vibrating system

ξ x :

Damping coefficient of the vibrating system

v :

Bifurcation parameter

v c :

The value of bifurcation

x 0 :

Initial displacement of vibrating body and material

\({\dot x_0},{\dot x_{10}}\) :

Initial velocity of vibrating body and material

λ x, r m, η12 :

Dimensionless parameters

\({\dot x_ - },{\dot x_{1 - }}\) :

Instantaneous velocity of vibrating body and material before the collision

\({\dot x_ + },{\dot x_{1 + }}\) :

Instantaneous velocity of vibrating body and material after the collision

a, b, C 1, C 2 :

Constants related to the initial conditions

\(\tilde a,\tilde b,{\tilde C_1},{\tilde C_2}\) :

Transient response coefficient

F 02[ΔX 2]:

The sum of quadratic terms about ΔX

F 03[ΔX 3]:

The sum of cubical terms about ΔX

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Acknowledgments

This study is supported by the National Natural Science Foundation of China (Grant No. 51705437); the Chinese Postdoctoral Fund (Grant No.2019M653482); Chengdu International Science and Technology Cooperation Project [Grant No. 2019-GH02-00035-HZ]. Sichuan Science and Technology Program [Grant No.2021JDRC0093].

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Authors and Affiliations

  1. School of Mechatronics Engineering, Southwest Petroleum University, Chengdu, China

    Yongjun Hou, Guang Xiong, Pan Fang, Yuwen Wang & Mingjun Du

  2. Key Laboratory of Oil & Gas Equipment, Ministry of Education, Southwest Petroleum University, Chengdu, China

    Yongjun Hou, Guang Xiong, Pan Fang, Yuwen Wang & Mingjun Du

Authors
  1. Yongjun Hou
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  2. Guang Xiong
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  4. Yuwen Wang
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Corresponding author

Correspondence to Yongjun Hou.

Additional information

Yongjun Hou completed his Ph.D. in Mechanical Design and Theory from Southwest Petroleum University, China, in 2002. He is a currently a Professor, a doctoral supervisor, and an excellent expert with outstanding contributions in Sichuan province. He has been engaged in mechanical design and theory, oil field machinery and other fields of teaching and research, focusing on mechanical vibration, mechanical and electrical system dynamics and applications.

Guang Xiong is currently pursuing the M.S. degree with Southwest Petroleum University, China. His research interests include vibration in mechanics, nonlinear systems, synchronization analysis, and numerical simulation.

Pan Fang completed his Ph.D. in Mechanical Engineering from Southwest Petroleum University, China, in 2016. Presently he is a Master Tutor at Southwest Petroleum University, China. His research interests include dynamics of multi-body systems and vibration control.

Yuwen Wang completed his M.S. degree in Mechanical Engineering form Southwest Petroleum University, China, in 2013. Presently he is a Lecturer at Southwest Petroleum University, China. His research interests include dynamics of multi-body systems and mechanical vibration.

Mingjun Du received the M.S. from Southwest Petroleum University, China, in 2017. He is currently pursuing the Ph.D. at Southwest Petroleum University, China. His research interests include dynamics of mechanical systems and nonlinear systems, and dynamics of synchronization systems.

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Hou, Y., Xiong, G., Fang, P. et al. Study on stability of self-synchronous far-resonant vibrating system of two eccentric rotors considering material impact. J Mech Sci Technol 35, 3271–3279 (2021). https://doi.org/10.1007/s12206-021-0701-2

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  • DOI: https://doi.org/10.1007/s12206-021-0701-2

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