Chaotic Motions in Resonant Separatrix Bands of a Parametrically Excited Pendulum

2000 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Numerical Results ◽  
Numerical Prediction ◽  
Chaotic Motions ◽  
Parametrically Excited ◽  
Spectrum Method ◽  
Energy Increment

Abstract The analytical conditions for the presence of primary (2M:1)-librational and (M:1)-rotational, resonant bands in a parametrically excited pendulum are obtained. The energy increment spectrum method is also developed for the numerical prediction of a specific primary resonant band. Illustrations of the analytical and numerical results for the appearance and destruction of the resonant bands are given for a comparison.

RESONANT LAYERS IN A PARAMETRICALLY EXCITED PENDULUM

2002 ◽  
Vol 12 (02) ◽  
pp. 409-419 ◽  
Author(s):  
ALBERT C. J. LUO
Keyword(s):  
Hamiltonian Systems ◽  
Analytical Method ◽  
Numerical Approach ◽  
Mapping Method ◽  
Numerical Results ◽  
Numerical Prediction ◽  
Parametrically Excited ◽  
Spectrum Method ◽  
Poincare Mapping ◽  
Energy Increment

The energy increment spectrum method is developed for the numerical prediction of a specific primary resonant layer, and the width of the resonant layer can be estimated through the energy increment spectrum. This numerical approach is applied to investigate the (2M:1)-librational and (M:1)-rotational, resonant layers in a parametrically excited pendulum, and the corresponding analytical conditions for such resonant layers are developed. The numerical approach predicts the appearance and disappearance of resonant layers in nonlinear Hamiltonian systems rather than the conventional Poincaré mapping method. Illustrations of the analytical and numerical results for the appearance and disappearance of the resonant layers are given. The width of the resonant layers in the paremetric pendulum is computed. The analytical method should be further improved through renormalization.

Numerical Prediction of Buckling in Ship Panel Structures

2007 ◽  
Vol 23 (03) ◽  
pp. 171-179
Author(s):  
Gonghyun Jung ◽  
T. D. Huang ◽  
Pingsha Dongg ◽  
Randall M. Dull ◽  
Christopher C. Conrardy ◽  
...  
Keyword(s):  
Shell Element ◽  
Numerical Results ◽  
Numerical Prediction ◽  
Test Panel ◽  
Physical Test ◽  
Thermal Tensioning ◽  
Transient Thermal

A shell-element-based numerical module was used to effectively predict welding-induced distortions. The numerical results were validated by comparison with a series of physical test panels. Eigenvalue analyses were performed to evaluate the buckling propensity of each test panel with and without transient thermal tensioning.

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.

Chaotic dynamics for a class of single-machine-infinite bus power system

2016 ◽  
Vol 24 (3) ◽  
pp. 582-587 ◽  
Author(s):  
Liangqiang Zhou ◽  
Fangqi Chen
Keyword(s):  
Power Systems ◽  
Power System ◽  
Single Machine ◽  
Chaotic Dynamics ◽  
Numerical Results ◽  
Chaotic Motions ◽  
System Parameters ◽  
Critical Curves ◽  
Single Machine Infinite Bus

The chaotic motions are investigated both analytically and numerically for a class of single-machine-infinite bus power systems. The mechanism and parametric conditions for chaotic motions of this system are obtained rigorously. The critical curves separating the chaotic and non-chaotic regions are presented. The chaotic feature of the system parameters is discussed in detail. It is shown that there exist chaotic bands for this system, and the bands vary with the system parameters. Some new dynamical phenomena are presented. Numerical results are given, which verify the analytical ones.

Chaotic Motions of a Parametrically Excited Pendulum With Non-Exponentially Small Parameters

2005 ◽  
Author(s):  
Chunqing Lu
Keyword(s):  
Poincaré Map ◽  
Poincare Map ◽  
Chaotic Motions ◽  
Parametrically Excited ◽  
Small Parameters ◽  
Exponentially Small

The paper mathematically proves that a pendulum with oscillatory forcing makes chaotic motions for certain parameters. The method is more intuitive than using the Poincare’ map. It provides more information about when the chaos occurs.

CFD Prediction of Smoke Spread During Hot-Smoke Tests in a Warehouse

2002 ◽  
Author(s):  
Seyit A. Ayranci ◽  
O¨zden F. Turan ◽  
Peter Taylor
Keyword(s):  
Satisfactory Agreement ◽  
Fire Safety ◽  
Safety Assessment ◽  
Numerical Results ◽  
Numerical Prediction ◽  
Smoke Spread ◽  
Fire Codes ◽  
Fire Experiments

A series of hot smoke tests is underway in a warehouse for comparison with more realistic fire experiments. Hot smoke tests are used commonly in the fire safety assessment of large enclosures. COMPACT is used in the numerical prediction of the spread of smoke and hot gasses in the warehouse during hot smoke tests. The comparison of numerical results with the measurements indicate satisfactory agreement. COMPACT will be utilized next in the prediction of fire experiments, as a contribution to the use of CFD tools in the development of performance based fire codes.

Chaotic Vibrations in Multi-Mass Discrete-Continuous Systems Torsionally Deformed with Local Nonlinearities

2014 ◽  
Vol 706 ◽  
pp. 6-13
Author(s):  
Amalia Pielorz ◽  
Danuta Sado
Keyword(s):  
Nonlinear Vibrations ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
Numerical Results ◽  
Continuous Systems ◽  
Chaotic Motions ◽  
Chaotic Vibrations ◽  
Wave Approach ◽  
Escape Phenomenon ◽  
External Moment

The paper deals with nonlinear vibrations in discrete-continuous mechanical systems consisting of rigid bodies connected by shafts torsionally deformed with local nonlinearities having hard or soft characteristics. The systems are loaded by an external moment harmonically changing in time. In the study the wave approach is used. Numerical results are presented for three-mass systems. In the study of regular vibrations in the case of a hard characteristic amplitude jumps are observed while in the case of a soft characteristic an escape phenomenon is observed. Irregular vibrations, including chaotic motions, are found for selected parameters of the systems.

Chaotic Motions of a Damped and Driven Morse Oscillator

2013 ◽  
Vol 459 ◽  
pp. 505-510
Author(s):  
Liang Qiang Zhou ◽  
Fang Qi Chen
Keyword(s):  
Numerical Methods ◽  
Dissociation Energy ◽  
Melnikov Method ◽  
Critical Value ◽  
Numerical Results ◽  
Morse Oscillator ◽  
Chaotic Motions ◽  
Critical Curves

With the Melnikov method and numerical methods, this paper investigate the chaotic motions of a damped and driven morse oscillator. The critical curves separating the chaotic and non-chaotic regions are obtained, which demonstrate that when the Morse spectroscopic term is fixed, for the case of large values of the period of the excitation, the critical value for chaotic motions decreases as the dissociation energy increases; while for the case of small values of the period of the excitation, the critical value for chaotic motions increases as the dissociation energy increases. It is also shown that when the dissociation energy is fixed, the critical value for chaotic motions always increase as the dissociation energy increases for any value of the period of the excitation. Some new dynamical phenomena are presented for this model. Numerical results verify the analytical ones.

scholarly journals The damping term makes the Smale-horseshoe heteroclinic chaotic motion easier

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Huijing Sun ◽  
Hongjun Cao
Keyword(s):  
Chaotic Motion ◽  
Analytical Techniques ◽  
Nonlinear Damping ◽  
Deterministic System ◽  
Maximum Lyapunov Exponent ◽  
Periodic Motions ◽  
Damping Term ◽  
Chaotic Motions ◽  
Parametrically Excited ◽  
Smale Horseshoe

<p style='text-indent:20px;'>The nonlinear Rayleigh damping term that is introduced to the classical parametrically excited pendulum makes the parametrically excited pendulum more complex and interesting. The effect of the nonlinear damping term on the new excitable systems is investigated based on analytical techniques such as Melnikov theory. The threshold conditions for the occurrence of Smale-horseshoe chaos of this deterministic system are obtained. Compared with the existing conclusion, i.e. the smaller the damping term is, the easier the chaotic motions become when the damping term is linear, our analysis, however, finds that the smaller or the larger the damping term is, the easier the Smale-horseshoe heteroclinic chaotic motions become. Moreover, the bifurcation diagram and the patterns of attractors in Poincaré map are studied carefully. The results demonstrate the new system exhibits rich dynamical phenomena: periodic motions, quasi-periodic motions and even chaotic motions. Importantly, according to the property of transitive as well as the fractal layers for a chaotic attractor, we can verify whether a attractor is a quasi-periodic one or a chaotic one when the maximum lyapunov exponent method is difficult to distinguish. Numerical simulations confirm the analytical predictions and show that the transition from regular to chaotic motion.</p>

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