Nonlinear Dynamics of Composite Laminated Circular Cylindrical Shell With Membranes in Thermal Environment

2018 ◽  
Author(s):  
Tao Liu ◽  
Wei Zhang ◽  
Yufei Zhang ◽  
Qian Wang
Keyword(s):  
Cylindrical Shell ◽  
Multiple Scales ◽  
Thermal Environment ◽  
Initial Conditions ◽  
Circular Cylindrical Shell ◽  
Period Doubling ◽  
Phase Portraits ◽  
Chaotic Motions ◽  
Temperature Parameter ◽  
Intermittent Chaos

This paper is focused on the chaotic dynamics of a composite laminated circular cylindrical shell with radially pre-stretched membranes at both ends and clamped along a generatrix. Based on the two-degree-of-freedom non-autonomous nonlinear equations of this system, the method of multiple scales is employed to obtain the four-dimensional nonlinear averaged equation. The resonant case considered here is the primary parametric resonance-1/2 subharmonic resonance and 1:1 internal resonance. Corresponding to several selected parameters, the periodic and chaotic motions of the composite laminated circular cylindrical shell clamped along a generatrix are demonstrated by the bifurcation diagrams, the maximum Lyapunov exponents, the phase portraits, the waveforms, the power spectrums and the Poincaré map. The temperature parameter excitation shows that the Pomeau-Manneville type intermittent chaos occur under the certain initial conditions. It is also found that there exist the twin phenomena between the Pomeau-Manneville type intermittent chaos and the period-doubling bifurcation.

Nonlinear Breathing Vibrations and Chaos of a Circular Truss Antenna with 1:2 Internal Resonance

2016 ◽  
Vol 26 (05) ◽  
pp. 1650077 ◽  
Author(s):  
W. Zhang ◽  
J. Chen ◽  
Y. Sun
Keyword(s):  
Differential Equation ◽  
Cylindrical Shell ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Thermal Environment ◽  
Circular Cylindrical Shell ◽  
Numerical Results ◽  
Thermal Excitation ◽  
Response Curves ◽  
Chaotic Motions

This paper investigates the nonlinear breathing vibrations and chaos of a circular truss antenna under changing thermal environment with 1:2 internal resonance for the first time. A continuum circular cylindrical shell clamped by one beam along its axial direction on one side is proposed to replace the circular truss antenna composed of the repetitive beam-like lattice by the principle of equivalent effect. The effective stiffness coefficients of the equivalent circular cylindrical shell are obtained. Based on the first-order shear deformation shell theory and the Hamilton’s principle, the nonlinear governing equations of motion are derived for the equivalent circular cylindrical shell. The Galerkin approach is utilized to discretize the nonlinear partial governing differential equation of motion to the ordinary differential equation for the equivalent circular cylindrical shell. The case of the 1:2 internal resonance, primary parametric resonance and 1/2 subharmonic resonance is taken into account. The method of multiple scales is used to obtain the four-dimensional averaged equation. The frequency-response curves and force-response curves are obtained when considering the strongly coupled of two modes. The numerical results indicate that there are the hardening type and softening type nonlinearities for the circular truss antenna. Numerical simulation is used to investigate the influences of the thermal excitation on the nonlinear breathing vibrations of the circular truss antenna. It is demonstrated from the numerical results that there exist the bifurcation and chaotic motions of the circular truss antenna.

Multi-Pulse Chaotic Dynamics of Eccentric Rotating Composite Laminated Circular Cylindrical Shell by Using the Extended Melnikov Method

2019 ◽  
Author(s):  
Yan Zheng ◽  
Wei Zhang ◽  
Tao Liu
Keyword(s):  
Cylindrical Shell ◽  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Circular Cylindrical Shell ◽  
Melnikov Method ◽  
Global Dynamics ◽  
Chaotic Motions ◽  
Extended Melnikov Method ◽  
Averaged Equations

Abstract The researches of global bifurcations and chaotic dynamics for high-dimensional nonlinear systems are extremely challenging. In this paper, we study the multi-pulse orbits and chaotic dynamics of an eccentric rotating composite laminated circular cylindrical shell. The four-dimensional averaged equations are obtained by directly using the multiple scales method under the case of the 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. The system is transformed to the averaged equations. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on the normal form obtained, the extended Melnikov method is utilized to analyze the multi-pulse global homoclinic bifurcations and chaotic dynamics for the eccentric rotating composite laminated circular cylindrical shell. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. From the averaged equations obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the eccentric rotating composite laminated circular cylindrical shell are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for the eccentric rotating composite laminated circular cylindrical shell.

Behavior of a Self-Sustained Electromechanical Transducer and Routes to Chaos

2005 ◽  
Vol 128 (3) ◽  
pp. 282-293 ◽  
Author(s):  
J. C. Chedjou ◽  
K. Kyamakya ◽  
I. Moussa ◽  
H.-P. Kuchenbecker ◽  
W. Mathis
Keyword(s):  
Electronic Circuit ◽  
Initial Conditions ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Electromechanical System ◽  
Phase Portraits ◽  
Coupling Coefficients ◽  
Oscillatory Solutions ◽  
Electromechanical Transducer ◽  
Design Engineers

This paper studies the dynamics of a self-sustained electromechanical transducer. The stability of fixed points in the linear response is examined. Their local bifurcations are investigated and different types of bifurcation likely to occur are found. Conditions for the occurrence of Hopf bifurcations are derived. Harmonic oscillatory solutions are obtained in both nonresonant and resonant cases. Their stability is analyzed in the resonant case. Various bifurcation diagrams associated to the largest one-dimensional (1-D) numerical Lyapunov exponent are obtained, and it is found that chaos can appear suddenly, through period doubling, period adding, or torus breakdown. The extreme sensitivity of the electromechanical system to both initial conditions and tiny variations of the coupling coefficients is also outlined. The experimental study of̱the electromechanical system is carried out. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the electromechanical system. Correspondences are established between the coefficients of the electromechanical system model and the components of the electronic circuit. Harmonic oscillatory solutions and phase portraits are obtained experimentally. One of the most important contributions of this work is to provide a set of reliable analytical expressions (formulas) describing the electromechanical system behavior. These formulas are of great importance for design engineers as they can be used to predict the states of the electromechanical systems and respectively to avoid their destruction. The reliability of the analytical formulas is demonstrated by the very good agreement with the results obtained by both the numeric and the experimental analysis.

scholarly journals Implementation on Electronic Circuits and RTR Pragmatical Adaptive Synchronization: Time-Reversed Uncertain Dynamical Systems' Analysis and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Shih-Yu Li ◽  
Cheng-Hsiung Yang ◽  
Li-Wei Ko ◽  
Chin-Teng Lin ◽  
Zheng-Ming Ge
Keyword(s):  
Adaptive Control ◽  
Stability Theory ◽  
Initial Conditions ◽  
Stability Theorem ◽  
Phase Portraits ◽  
Adaptive Synchronization ◽  
Complex Signal ◽  
Uncertain Parameters ◽  
Chaotic Motions ◽  
Lorentz System

We expose the chaotic attractors of time-reversed nonlinear system, further implement its behavior on electronic circuit, and apply the pragmatical asymptotically stability theory to strictly prove that the adaptive synchronization of given master and slave systems with uncertain parameters can be achieved. In this paper, the variety chaotic motions of time-reversed Lorentz system are investigated through Lyapunov exponents, phase portraits, and bifurcation diagrams. For further applying the complex signal in secure communication and file encryption, we construct the circuit to show the similar chaotic signal of time-reversed Lorentz system. In addition, pragmatical asymptotically stability theorem and an assumption of equal probability for ergodic initial conditions (Ge et al., 1999, Ge and Yu, 2000, and Matsushima, 1972) are proposed to strictly prove that adaptive control can be accomplished successfully. The current scheme of adaptive control—by traditional Lyapunov stability theorem and Barbalat lemma, which are used to prove the error vector—approaches zero, as time approaches infinity. However, the core question—why the estimated or given parameters also approach to the uncertain parameters—remains without answer. By the new stability theory, those estimated parameters can be proved approaching the uncertain values strictly, and the simulation results are shown in this paper.

Internal Resonance and Nonlinear Vibrations of Eccentric Rotating Composite Laminated Circular Cylindrical Shell

2019 ◽  
Author(s):  
Tao Liu ◽  
Wei Zhang ◽  
Yan Zheng ◽  
Yufei Zhang
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Shear Deformation Theory ◽  
Internal Resonances

Abstract This paper is focused on the internal resonances and nonlinear vibrations of an eccentric rotating composite laminated circular cylindrical shell subjected to the lateral excitation and the parametric excitation. Based on Love thin shear deformation theory, the nonlinear partial differential equations of motion for the eccentric rotating composite laminated circular cylindrical shell are established by Hamilton’s principle, which are derived into a set of coupled nonlinear ordinary differential equations by the Galerkin discretization. The excitation conditions of the internal resonance is found through the Campbell diagram, and the effects of eccentricity ratio and geometric papameters on the internal resonance of the eccentric rotating system are studied. Then, the method of multiple scales is employed to obtain the four-dimensional nonlinear averaged equations in the case of 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. Finally, we study the nonlinear vibrations of the eccentric rotating composite laminated circular cylindrical shell systems.

On Nonlinear Motions of Two-Degree-of-Freedom Nonlinear Systems with Repeated Linearized Natural Frequencies

2019 ◽  
Vol 29 (10) ◽  
pp. 1950132
Author(s):  
Hua-Zhen An ◽  
Xiao-Dong Yang ◽  
Feng Liang ◽  
Wei Zhang ◽  
Tian-Zhi Yang ◽  
...  
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Phase Portraits ◽  
Chaotic Motions ◽  
Nonlinear Parameters ◽  
Energy Ratios ◽  
Two Degree Of Freedom ◽  
Nonlinear Motions ◽  
Phase Differences

In this paper, we investigate systematically the vibration of a typical 2DOF nonlinear system with repeated linearized natural frequencies. By application of Descartes’ rule of signs, we demonstrate that there are 14 types of roots describing different modal motions for varying nonlinear parameters. The method of multiple scales is used to obtain the amplitude-phase portraits by introducing the energy ratios and phase differences. The typical nonlinear in-unison and elliptic out-of-unison modal motions are located for the 14 types of roots and then validated by numerical simulations. It is found that the normal in-unison modal motions, elliptic out-of-unison modal motions are analogous to the polarization of classical optic theory. Further, some kinds of periodic and chaotic motions under out-of-unison and in-unison excitations are investigated numerically. The result of this study offers a detailed discussion of nonlinear modal motions and responses of 2DOF systems with cubic nonlinear terms.

Multistability and Bubbling Route to Chaos in a Deterministic Model for Geomagnetic Field Reversals

2019 ◽  
Vol 29 (12) ◽  
pp. 1930034
Author(s):  
Paulo C. Rech ◽  
Sudarshan Dhua ◽  
N. C. Pati
Keyword(s):  
Fixed Point ◽  
Geomagnetic Field ◽  
Deterministic Model ◽  
Initial Conditions ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Multiple Attractors ◽  
System A ◽  
Two Parameters

We report coexisting multiple attractors and birth of chaos via period-bubbling cascades in a model of geomagnetic field reversals. The model system comprises a set of three coupled first-order quadratic nonlinear equations with three control parameters. Up to seven kinds of multistable attractors, viz. fixed point-periodic, fixed point-chaotic, periodic–periodic, periodic-chaotic, chaotic–chaotic, fixed point-periodic–periodic, fixed point-periodic-chaotic are obtained depending on the initial conditions for critical parameter sets. Antimonotonicity is a striking characteristic feature of nonlinear systems through which a full Feigenbaum tree corresponding to creation and annihilation of period-doubling cascades is developed. By analyzing the two-parameters dependent dynamics of the system, a critical biparameter zone is identified, where antimonotonicity comes into existence. The complex dynamical behaviors of the system are explored using phase portraits, bifurcation diagrams, Lyapunov exponents, isoperiodic diagram, and basins of attraction.

Periodic and Chaotic Oscillations of Laminated Composite Piezoelectric Rectangular Plate With 1:3 Internal Resonance

2007 ◽  
Author(s):  
Wei Zhang ◽  
Li-Hua Chen ◽  
Zhi-Gang Yao ◽  
Xiao-Li Yang
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Initial Conditions ◽  
Laminated Composite ◽  
Rectangular Plates ◽  
Third Order ◽  
Chaotic Motions ◽  
Transverse Loads ◽  
Averaged Equations ◽  
Laminate Theory

The chaotic dynamics of parametrically excited, simply supported laminated composite piezoelectric rectangular plates are analyzed, The plates are forced by transverse loads. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded in them. Firstly, based on von Karman-type equations and third-order shear deformation laminate theory of Reddy, the nonlinear equations of motions of the laminated composite piezoelectric rectangular plates are derived. Here, we consider the piezoelectric parametric loads and in-plane parametric loads acting in both x-direction and y-direction. Then, the Galerkin’s approach is applied to convert partial differential equations to the ordinary differential equations. The method of multiple scales is used to obtain the averaged equations. Finally, based on the averaged equations, periodic and chaotic motions of the plates are found by using numerical simulation. The numerical results show the existence of periodic and chaotic motions in averaged equations. The chaotic responses are sensitive to initial conditions especially to forcing loads and the parametric excitation.

Auto and Cross-Bispectral Analysis of a System of Two Coupled Oscillators With Quadratic Nonlinearities Possessing Chaotic Motion

1992 ◽  
Vol 59 (3) ◽  
pp. 657-663 ◽  
Author(s):  
Charles Pezeshki ◽  
Steve Elgar ◽  
R. Krishna ◽  
T. D. Burton
Keyword(s):  
Degrees Of Freedom ◽  
Multiple Scales ◽  
Period Doubling ◽  
Phase Coupling ◽  
Chaotic Motions ◽  
First Order ◽  
Yield Information ◽  
Quadratic Nonlinearities ◽  
Route To Chaos ◽  
The Individual

Auto and cross-bispectral analyses of a two-degree-of-freedom system with quadratic nonlinearities having two-to-one internal (autoparametric) resonance are presented. Following the work of Nayfeh (1987), the method of multiple scales is used to obtain a first-order uniform expansion yielding four first-order nonlinear ordinary differential equations governing the modulation of the amplitudes and phases of the two modes. The particular case of parametric resonance of the first mode considered in this paper admits Hopf bifurcations and a pure period doubling route to chaos. Auto bicoherence spectra isolate the phase coupling between increasing numbers of triads of Fourier components for a pure period doubling route to chaos for the individual degrees-of freedom. Cross-bicoherence spectra, on the other hand, yield information about the phase coupling between the two degrees-of-freedom. The results presented here confirm the capacity of bispectral techniques to identify a quadratically nonlinear mechanical system that possesses chaotic motions. For the chaotic case, cross-bicoherence spectra indicate that most of the nonlinear energy transfer between the modes is owing to cross-coupling between phase modulations rather than between amplitude modulations.

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