scholarly journals Periodic Motion and Transition of a Vibro‐Impact System with Multilevel Elastic Constraints

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Jie Ding ◽  
Chao Wang ◽  
Wangcai Ding
Keyword(s):  
Periodic Motion ◽  
Simulation Method ◽  
Scientific Basis ◽  
Mapping Method ◽  
Periodic Motions ◽  
Coexistence Of Attractors ◽  
Vibroimpact System ◽  
The Stability ◽  
Elastic Constraints ◽  
Lyapunov Exponent Spectrum

In this paper, a single-degree-of-freedom vibroimpact system with multilevel elastic constraints is taken as the research object. By constructing the Poincaré map of the system and calculating the Lyapunov exponent spectrum of the system, the stability of the system is determined. Using the multiparameter collaborative numerical simulation method, the parameter domains of various periodic motions are determined, and the diversity and transition characteristics of periodic motions are revealed. At the same time, combined with the cell mapping method, the coexistence of attractors induced due to grazing bifurcation, saddle-node bifurcation, and boundary crisis is studied. Finally, the influence of system parameters on periodic motion distribution is analyzed, which provides a scientific basis for system parameter optimization.

On an Independent Period-1 Motion in a Flexible Rotor System

2019 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C. J. Luo
Keyword(s):  
Numerical Simulations ◽  
Internal Resonance ◽  
Mapping Method ◽  
Rotor System ◽  
Flexible Rotor ◽  
Periodic Motions ◽  
Phase Trajectories ◽  
Stability And Bifurcation ◽  
The Stability ◽  
Nonlinear Rotor System

Abstract This paper investigates stable and unstable period-1 motions in a rotor system through the discrete mapping method. The discrete mapping of a nonlinear rotor system is for stable and unstable period-1 motions. The stability and bifurcation of periodic motions are determined. Numerical simulations of periodic motions are completed and phase trajectories, displacement orbits and velocity plane are illustrated. The period-1 motion near the internal resonance is determined with large vibration in the nonlinear rotor system.

Nonlinear Dynamic Analysis of a Flexible Rod Fastening Rotor Bearing System

2010 ◽  
Author(s):  
Heng Liu ◽  
Chen Li ◽  
Weimin Wang ◽  
Xiaobin Qi ◽  
Minqing Jing
Keyword(s):  
Periodic Motion ◽  
Great Influence ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mass Eccentricity ◽  
Rotor Bearing ◽  
Rod Fastening Rotor ◽  
The Stability ◽  
Tie Rods ◽  
Bearing System

This paper is concerned the stability and bifurcation of a flexible rod-fastening rotor bearing system (FRRBS). Here the shaft is considered as an integral or continuous structure and be modeled by using Timoshenko beam-shaft element which can take the effects of axial load into consideration. And using Hamilton’s principle, model tie rods distributed along the circumference as a constant stiffness matrix and an add-moment which caused by unbalanced pre-tightening forces. Then the model is reduced by a component mode synthesis method, which can conveniently account for nonlinear oil film forces of the bearing. This study focuses on the influence of nonlinearities on the stability and bifurcation of T periodic motion of the FRRBS subjected to the influence of mass eccentricity. The periodic motions and their stability margin are obtained by shooting method and path-following technique. The local stability and bifurcation behaviors of periodic motions are obtained by Floquet theory. The results indicate that mass eccentricity and unbalanced pre-tightening forces of tie rods have great influence on nonlinear stability and bifurcation of the T periodic motion of system, cause the spillover of system nonlinear dynamics and degradation of stability and bifurcation of T periodic motion.

Period-m Motions in a Periodically Forced van der Pol Oscillator

2013 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Fourier Series ◽  
Periodic Motion ◽  
Approximate Solutions ◽  
Van Der Pol Oscillator ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Stable Period ◽  
Stability And Bifurcation Analysis ◽  
Series Expression ◽  
The Stability

Period-m motions in a periodically forced, van der Pol oscillator are investigated through the Fourier series expression, and the stability and bifurcation analysis of such periodic motions are carried out. To verify the approximate solutions of period-m motions, numerical illustrations are given. Period-m motions are separated by quasi-periodic motion or chaos, and the stable period-m motions are in independent periodic motion windows.

Effect of Thrust Magnetic Bearing on Stability and Bifurcation of a Flexible Rotor Active Magnetic Bearing System

2003 ◽  
Vol 125 (3) ◽  
pp. 307-316 ◽  
Author(s):  
Y. S. Ho ◽  
H. Liu ◽  
L. Yu
Keyword(s):  
Periodic Motion ◽  
Magnetic Bearing ◽  
Rotor System ◽  
Magnetic Bearings ◽  
Active Magnetic Bearing ◽  
Flexible Rotor ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mass Eccentricity ◽  
The Stability

This paper is concerned with the effect of a thrust active magnetic bearing (TAMB) on the stability and bifurcation of an active magnetic bearing rotor system (AMBRS). The shaft is flexible and modeled by using the finite element method that can take the effects of inertia and shear into consideration. The model is reduced by a component mode synthesis method, which can conveniently account for nonlinear magnetic forces and moments of the bearing. Then the system equations are obtained by combining the equations of the reduced mechanical system and the equations of the decentralized PID controllers. This study focuses on the influence of nonlinearities on the stability and bifurcation of T periodic motion of the AMBRS subjected to the influences of both journal and thrust active magnetic bearings and mass eccentricity simultaneously. In the stability analysis, only periodic motion is investigated. The periodic motions and their stability margins are obtained by using shooting method and path-following technique. The local stability and bifurcation behaviors of periodic motions are obtained by using Floquet theory. The results indicate that the TAMB and mass eccentricity have great influence on nonlinear stability and bifurcation of the T periodic motion of system, cause the spillover of system nonlinear dynamics and degradation of stability and bifurcation of T periodic motion. Therefore, sufficient attention should be paid to these factors in the analysis and design of a flexible rotor system equipped with both journal and thrust magnetic bearings in order to ensure system reliability.

scholarly journals Double Grazing Periodic Motions and Bifurcations in a Vibroimpact System with Bilateral Stops

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Qunhong Li ◽  
Limei Wei ◽  
Jieyan Tan ◽  
Jiezhen Xi
Keyword(s):  
Periodic Orbit ◽  
Periodic Motion ◽  
Poincaré Map ◽  
Complex Form ◽  
Original System ◽  
Existence Condition ◽  
Poincare Map ◽  
Periodic Motions ◽  
Continuous Transition ◽  
Vibroimpact System

The double grazing periodic motions and bifurcations are investigated for a two-degree-of-freedom vibroimpact system with symmetrical rigid stops in this paper. From the initial condition and periodicity, existence of the double grazing periodic motion of the system is discussed. Using the existence condition derived, a set of parameter values is found that generates a double grazing periodic motion in the considered system. By extending the discontinuity mapping of one constraint surface to that of two constraint surfaces, the Poincaré map of the vibroimpact system is constructed in the proximity of the grazing point of a double grazing periodic orbit, which has a more complex form than that of the single grazing periodic orbit. The grazing bifurcation of the system is analyzed through the Poincaré map with clearance as a bifurcation parameter. Numerical simulations show that there is a continuous transition from the chaotic band to a period-1 periodic motion, which is confirmed by the numerical simulation of the original system.

Periodic-Doubling and Hopf Bifurcations of a Two-Degree-of-Freedom System with Elastic Constraints and Clearances

2014 ◽  
Vol 1006-1007 ◽  
pp. 285-289
Author(s):  
Xi Feng Zhu ◽  
Quan Fu Gao
Keyword(s):  
Hopf Bifurcation ◽  
Period Doubling ◽  
Simulation Method ◽  
Degree Of Freedom ◽  
Local Bifurcations ◽  
Stiffness And Damping Coefficient ◽  
The Stability ◽  
Elastic Constraints ◽  
Stiffness And Damping ◽  
Two Degree Of Freedom

Based on the study of a dual component system with elastic constraints, the stability and local bifurcations of the soft-impacts system, such as piecewise property and singularity, was analyzed by using the Poincaré map and Runge-Kutta numerical simulation method. The routes from periodic motions to chaos, via Hopf bifurcation and period-doubling bifurcation, were investigated exactly. In the large constraint stiffness case, the period-doubling and Hopf bifurcation exist in the two-degree-of-freedom system with elastic constraints and clearances. The clearances of the system, stiffness and damping coefficient of the elastic constraints is the main reasons for influencing the chaotic motion. The steady 1-1-1 period orbits or 2-1-1 period orbits will exist within a wideband frequency range and the value of velocity will be higher when appropriate system parameters are chosen.

Existence, Stability and Bifurcation of Periodic Motions in a Periodically Forced, Piecewise, Linear Oscillator With Impacts

2004 ◽  
Author(s):  
Albert C. J. Luo ◽  
Lidi Chen
Keyword(s):  
Nonlinear Dynamics ◽  
Periodic Motion ◽  
Piecewise Linear ◽  
Linear Oscillator ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Perfectly Plastic ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
The Stability

The nonlinear dynamics of a generalized, piecewise linear oscillator with perfectly plastic impacts is investigated. The generic mappings based on the discontinuous boundaries are constructed. Furthermore, the mapping structures are developed for the analytical prediction of periodic motions of such a system. The stability and bifurcation conditions for specified periodic motions are obtained. The periodic motions and grazing motion are demonstrated. This model is applicable to prediction of periodic motion in nonlinear dynamics of gear transmission systems.

Global Chaos in a Periodically Forced, Linear System With a Dead-Zone Restoring Force

2003 ◽  
Author(s):  
Albert C. J. Luo ◽  
Santhosh Menon
Keyword(s):  
Linear System ◽  
Periodic Motion ◽  
Dead Zone ◽  
Poincaré Mapping ◽  
Periodic Motions ◽  
Restoring Force ◽  
Chaotic Motions ◽  
Mapping Structures ◽  
Poincare Mapping ◽  
The Stability

The Poincare mapping and the corresponding mapping sections for global motions in a linear system possessing a dead-zone restoring force are developed through the switching planes pertaining to the two constraints. The global periodic motions based on the Poincare mapping are determined, and the analysis for the stability and bifurcation of periodic motion is carried out. From the global periodic motions, the global chaos in such a system is investigated numerically. The bifurcation scenario with varying parameters was presented. The mapping structures of periodic and chaotic motions are discussed. The Poincare mapping sections for global chaos are given for illustration. The grazing phenomenon embedded in chaotic motion is observed.

Investigation of the stability of periodic motions in a nonlinear automatic control system based on the fast fourier transform

2003 ◽  
Vol 3 ◽  
pp. 297-307
Author(s):  
V.V. Denisov
Keyword(s):  
Fourier Transform ◽  
Control System ◽  
Automatic Control ◽  
Fast Fourier Transform ◽  
Automatic Control System ◽  
Resonance Phenomenon ◽  
Periodic Motions ◽  
Multiply Connected ◽  
Non Linear ◽  
The Stability

An approach to the study of the stability of non-linear multiply connected systems of automatic control by means of a fast Fourier transform and the resonance phenomenon is considered.

Stability Prediction of Surrounding Rocks and Optimum Design of Roof-Bolt Parameters in Deep Roadway

2013 ◽  
Vol 353-356 ◽  
pp. 436-439
Author(s):  
De Sen Kong ◽  
Yong Po Chen
Keyword(s):  
Prediction Method ◽  
Simulation Method ◽  
Complex Stress ◽  
Roof Bolt ◽  
Deformation And Failure ◽  
Long Run ◽  
Research Findings ◽  
Stress Environment ◽  
Classification Prediction ◽  
The Stability

In order to forecast the stability of deep roadway and optimize the parameters of bolts, the complex stress environment and the multivariate surrounding rocks characteristics of deep roadway were analyzed. Then the classification prediction method and the numerical simulation method were simultaneously used to analysis the stability of surrounding rocks. Furthermore, the supporting parameters of bolts were also designed optimally. It was shown that the characteristics of stress distribution, deformation and failure zone of surrounding rocks are not ideal. So it is necessary to optimize the supporting parameters of deep roadway. All these research findings will provide the theory basis for bolts of deep roadway and will ensure the optimization of bolts and the stability of deep roadway in the long run.

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