The Chaotic Behavior of Symmetric Laminated Composite Arch

2010 ◽  
Vol 29-32 ◽  
pp. 287-292
Author(s):  
Jian Jun Wang ◽  
Zhi Jun Han ◽  
Chao Kang ◽  
Guo Yun Lu ◽  
Shan Yuan Zhang
Keyword(s):  
Chaotic Motion ◽  
Chaotic Behavior ◽  
Steady Motion ◽  
Time History ◽  
Laminated Composite ◽  
Critical Conditions ◽  
Loading Frequency ◽  
Periodic Excitation ◽  
Chaotic Region ◽  
Nonlinear Dynamic Equations

Chaotic motion of symmetric laminated composite arch with two hinge supports under transverse periodic excitation was investigated. The nonlinear dynamic equations of the arch are changed into the square-order and cubic nonlinear differential dynamic system by Galerkin method, and its homoclinic orbit parameter equations are also acquired. The critical conditions of horseshoe-type chaos are obtained by using Melnikov function. The influence of loading frequency on chaotic region are analysed by numerical calculation. The motion behaviors of system are described through the bifurcation diagrams, the time-history curve, phase portrait and Poincaré map. The results are given as follows. The influence of loading frequency on chaotic region are significant. When the height of arch reach some value, the system can occur horseshoe-type chaos. The system of symmetric laminated composite arch under transverse periodic excitation may occur steady motion and chaotic motion.

scholarly journals Confusion threshold study of the Duffing oscillator with a nonlinear fractional damping term

2020 ◽  
pp. 146134842092268
Author(s):  
Wang Mei-Qi ◽  
Ma Wen-Li ◽  
Chen En-Li ◽  
Yang Shao-Pu ◽  
Chang Yu-Jian ◽  
...  
Keyword(s):  
Chaotic Motion ◽  
Numerical Solutions ◽  
Duffing Oscillator ◽  
Time History ◽  
Nonlinear Damping ◽  
Critical Conditions ◽  
Fractional Damping ◽  
Nonlinear Parameters ◽  
Smale Horseshoe ◽  
Melnikov Theory

In this study, the critical conditions for generating chaos in a Duffing oscillator with nonlinear damping and fractional derivative are investigated. The Melnikov function of the Duffing oscillator is established based on Melnikov theory. The necessary analytical conditions and critical value curves of chaotic motion in the sense of Smale horseshoe are obtained. The numerical solutions of chaotic motion, including time history diagram, frequency spectrum diagram, phase diagram, and Poincare map, are studied. The correctness of the analytical solution is verified through a comparison of numerical and analytical calculations. The effects of linear and nonlinear parameters on chaotic motion are also analyzed. These results are relevant to the study of system dynamics.

A case study in bifurcation theory

2017 ◽  
Vol 28 (08) ◽  
pp. 1750104 ◽  
Author(s):  
Youssef Khmou
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Theory ◽  
Chaotic Behavior ◽  
Numerical Study ◽  
Logistic Map ◽  
Logistic Function ◽  
Second Order ◽  
Short Paper ◽  
Chaotic Region

This short paper is focused on the bifurcation theory found in map functions called evolution functions that are used in dynamical systems. The most well-known example of discrete iterative function is the logistic map that puts into evidence bifurcation and chaotic behavior of the topology of the logistic function. We propose a new iterative function based on Lorentizan function and its generalized versions, based on numerical study, it is found that the bifurcation of the Lorentzian function is of second-order where it is characterized by the absence of chaotic region.

scholarly journals Analysis of Nonlinear Dynamic Characteristics of a Mechanical-Electromagnetic Vibration System with Rubbing

2019 ◽  
Vol 9 (21) ◽  
pp. 4612
Author(s):  
Yiming Li ◽  
Zhilong Huang ◽  
He Li ◽  
Guiqiu Song
Keyword(s):  
Friction Coefficient ◽  
Dynamic Characteristics ◽  
Chaotic Motion ◽  
Nonlinear Dynamic ◽  
Response Amplitude ◽  
Jump Discontinuity ◽  
Excitation Current ◽  
Radial Stiffness ◽  
Chaotic Region ◽  
Coupling System

In this study, a rotor-bearing-runner system (RBRS) considering multiple nonlinear factors is established, and the complex nonlinear dynamic behavior of the coupling system is studied. The effects of excitation current, radial stiffness, and friction coefficient on dynamic characteristics are analyzed by numerical simulation. The research results show that the dynamic properties of the coupling system caused by different nonlinear factors are interactional. With the changes of different parameters, the RBRS presents multiple motion states, including periodic-n, quasi-periodic, and chaotic motion. The increase of the excitation current Ij has a certain inhibitory effect on the response amplitude of the system and makes the motion state of the system more complex, the chaotic motion wider, and the jump discontinuity enhanced. With the increase of radial stiffness kr, the motion complexity of the coupling system increases, the chaotic region increases, the response amplitude increases, and the vibration intensity increases. With the increase of the friction coefficient μ, the chaotic region increases first and decreases, the different motions alternate frequently, and the response amplitude gradually increases. This study can not only help to understand the dynamic characteristics of RBRS, but also help the stable operation of the generator set.

scholarly journals Nonlinear Dynamic Analysis of Rotor Rub-Impact System

2019 ◽  
Vol 2019 ◽  
pp. 1-20
Author(s):  
Youfeng Zhu ◽  
Zibo Wang ◽  
Qiang Wang ◽  
Xinhua Liu ◽  
Hongyu Zang ◽  
...  
Keyword(s):  
Periodic Motion ◽  
Chaotic Motion ◽  
Nonlinear Dynamic Analysis ◽  
Period Doubling ◽  
Time History ◽  
Rotor System ◽  
Quasiperiodic Motion ◽  
Rotor Bearing ◽  
Motion Period ◽  
Bearing System

A dynamic model of a double-disk rub-impact rotor-bearing system with rubbing fault is established. The dynamic differential equation of the system is solved by combining the numerical integration method with MATLAB. And the influence of rotor speed, disc eccentricity, and stator stiffness on the response of the rotor-bearing system is analyzed. In the rotor system, the time history diagram, the axis locus diagram, the phase diagram, and the Poincaré section diagram in different rotational speeds are drawn. The characteristics of the periodic motion, quasiperiodic motion, and chaotic motion of the system in a given speed range are described in detail. The ways of the system entering and leaving chaos are revealed. The transformation and evolution process of the periodic motion, quasiperiodic motion, and chaotic motion are also analyzed. It shows that the rotor system enters chaos by the way of the period-doubling bifurcation. With the increase of the eccentricity, the quasi-periodicity evolution is chaotic. The quasiperiodic motion evolves into the periodic three motion phenomenon. And the increase of the stator stiffness will reduce the chaotic motion period.

Stability of a Mathematical Model with Piecewise Constant Arguments for Tumor-Immune Interaction Under Drug Therapy

2019 ◽  
Vol 29 (01) ◽  
pp. 1950009 ◽  
Author(s):  
Zonghong Feng ◽  
Xinxing Wu ◽  
Luo Yang
Keyword(s):  
Mathematical Model ◽  
Growth Rate ◽  
Drug Therapy ◽  
Tumor Growth ◽  
Tumor Cells ◽  
Chaotic Behavior ◽  
Sufficient Conditions ◽  
Tumor Growth Rate ◽  
Chaotic Region ◽  
Treatment Parameter

This paper studies a mathematical model for the interaction between tumor cells and Cytotoxic T lymphocytes (CTLs) under drug therapy. We obtain some sufficient conditions for the local and global asymptotical stabilities of the system by using Schur–Cohn criterion and the theory of Lyapunov function. In addition, it is known that the system without any treatment may undergo Neimark–Sacker bifurcation, and there may exist a chaotic region of values of tumor growth rate where the system exhibits chaotic behavior. So it is important to narrow the chaotic region. This may be done by increasing the intensity of the treatment to some extent. Moreover, for a fixed value of tumor growth rate in the chaotic region, a threshold value [Formula: see text] is predicted of the treatment parameter [Formula: see text]. We can see Neimark–Sacker bifurcation of the system when [Formula: see text], and the chaotic behavior for tumor cells ends and the system becomes locally asymptotically stable when [Formula: see text].

Chaotic Flexural Oscillations of Embedded Non-Local Nanotubes Subjected to Axial Harmonic Force

2017 ◽  
Author(s):  
Zia Saadatnia ◽  
Ebrahim Esmailzadeh
Keyword(s):  
Chaotic Behavior ◽  
Scale Effects ◽  
Nonlinear Differential Equations ◽  
Mathematical Formulation ◽  
Time History ◽  
Critical Parameters ◽  
Local Theory ◽  
Small Scale ◽  
External Excitation ◽  
Non Local

Chaotic behavior of an embedded carbon nanotube subjected to an external excitation and the combinational static-dynamic axial loads is investigated. Mathematical formulation has been developed based on the non-local theory in order to reflect the small-scale effects. The tube is supported by the Kelvin-Voigt viscoelastic foundation and the Galerkin method is utilized to solve the governing nonlinear differential equations. The vibration behavior of the system for the parameters of a real model is studied and different vibration responses of the nanotube such as the periodic, quasi-periodic and the chaotic behaviors are detected. The bifurcation diagrams for several critical parameters, including the amplitude of external excitation and the axial applied load are presented. The time history diagram, phase-plane trajectories, and the Poincaré map are presented as the three appropriate techniques for diagnosing the system behavior under various conditions.

Chaotic motion of a nonlinear near resonance centrifuge

2020 ◽  
Vol 51 (11) ◽  
pp. 189-194
Author(s):  
Feng Guo ◽  
Na Li
Keyword(s):  
Numerical Simulation ◽  
Equilibrium Point ◽  
Chaotic Motion ◽  
Melnikov Method ◽  
Melnikov Function ◽  
Critical Parameters ◽  
Critical Conditions ◽  
External Excitation ◽  
Chaotic Motions ◽  
Near Resonance

The equilibrium point and stability of the motion equation of the nonlinear near resonance centrifuge is studied, and the critical conditions for chaotic motions of the system under external excitation are studied by Melnikov method. The expression of Melnikov function and the boundary value between chaotic and non-chaotic regions are given. According to the range of parameters, the numerical simulations are carried out. The results show that the critical parameters of chaotic motion determined using Melnikov method are consistent with that obtained by the numerical simulation. This method effectively judges the occurrence of chaotic motion.

scholarly journals Comment on the Chaotic Behaviour of van der Pol Equations with an External Periodic Excitation

1989 ◽  
Vol 44 (2) ◽  
pp. 160-162
Author(s):  
W.-H. Steeb ◽  
Jeun Chyuan Huang ◽  
Yih Shun Gou
Keyword(s):  
Limit Cycle ◽  
Chaotic Motion ◽  
Chaotic Behaviour ◽  
Surprising Result ◽  
Periodic Excitation ◽  
Van Der Pol ◽  
Periodic Force ◽  
Cycle System ◽  
Van Der Pol Equations

Abstract The limit cycle system with an external periodic force d2u/dt2 - a( 1 - u2)du/dt + un = kcos(Ωf) (n = 1, 3, 5,...) can show chaotic behaviour for certain values of a, k and Ω. We study the influence of n on the chaotic behaviour. For n = 1 we select values which result in chaotic motion of the system. Then we investigate the behaviour of the system for n = 3, 5 and 7. Introducing the nonlinearity un(n - 3, 5, 7) gives the surprising result that the chaotic motion ceases to exist.

Chaos Control of Vehicle Nonlinear Suspension System with Multi-Frequency Excitations by Nonlinear Feedback

2011 ◽  
Vol 55-57 ◽  
pp. 1156-1161
Author(s):  
Jing Yue Wang ◽  
Hao Tian Wang ◽  
Li Min Zheng
Keyword(s):  
Degrees Of Freedom ◽  
Ride Comfort ◽  
Control Method ◽  
Chaotic Behavior ◽  
Time History ◽  
Suspension System ◽  
Vehicle Suspension ◽  
Nonlinear Feedback ◽  
Suspension Model ◽  
Vehicle Suspension System

Vehicle suspension system with hysteretic nonlinearity has obvious nonlinear characteristics, which directly cause the system to the possibility of existence of bifurcation and chaos. Two degrees of freedom for the 1/4 body suspension model is established and the behavior of the system under road multi-frequency excitations is analyzed. In the paper, it reveals the existence of chaos in the system with the Poincaré map, phase diagram, time history graph, and its chaotic behavior is controlled by nonlinear feedback. Numerical simulation shows the effectiveness and feasibility of the control method with improved ride comfort. The results may supply theoretical bases for the analysis and optimal design of the vehicle suspension system.

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