Periodic Temperature Responses in a Thermal System Under a Periodic Heating

2021 ◽  
Author(s):  
Bo Yu ◽  
Albert C. J. Luo
Keyword(s):  
Excitation Frequency ◽  
Mapping Method ◽  
Numerical Results ◽  
Eigenvalue Analysis ◽  
Thermal System ◽  
Periodic Heating ◽  
Periodic Response ◽  
Implicit Mapping ◽  
Periodic Temperature ◽  
Temperature Responses

Abstract In this paper, the periodic temperature responses of a thermal system under a periodic heating input are studied. Using the implicit mapping method, periodic temperature responses varying with excitation frequency are predicted for different input amplitudes. The corresponding stability of the periodic responses are discussed through eigenvalue analysis. The experimental and numerical results of the periodic response are presented for comparison to the analytical results.

Period-3 Motions in a Parametrically Exited Inverted Pendulum

2020 ◽  
Author(s):  
Albert C. J. Luo ◽  
Chuan Guo
Keyword(s):  
Numerical Simulations ◽  
Inverted Pendulum ◽  
Period Doubling ◽  
Mapping Method ◽  
Eigenvalue Analysis ◽  
Symmetric Motion ◽  
Bifurcation Tree ◽  
Implicit Mapping ◽  
Stability And Bifurcation ◽  
Period Doubling Bifurcations

Abstract In this paper, period-3 motions in a parametrically exited inverted pendulum are analytically investigated through a discrete implicit mapping method. The corresponding stability and bifurcation conditions of the period-3 motions are predicted through eigenvalue analysis. The symmetric and asymmetric period-3 motions are obtained on the bifurcation tree, and the period-doubling bifurcations of the asymmetric period-3 motions are observed. The saddle-node and Neimark bifurcations for symmetric period-3 motions are obtained. The saddle-bifurcations of the symmetric period-3 motions are for symmetric motion appearance (or vanishing) and onsets of asymmetric period-3 motion. Numerical simulations of the period-3 motions in the inverted pendulum are completed from analytical predictions for illustration of motion complexity and characteristics.

Period Motions in a Periodically Forced, Damped Double Pendulum

2018 ◽  
Author(s):  
Albert C. J. Luo ◽  
Chuan Guo
Keyword(s):  
Differential Equation ◽  
Numerical Simulation ◽  
Mapping Method ◽  
Eigenvalue Analysis ◽  
Double Pendulum ◽  
Implicit Mapping ◽  
Stability And Bifurcation

In this paper, period motions in a periodically forced, damped, double pendulum are analytically predicted through a discrete implicit mapping method. The implicit mapping is established via the discretized differential equation. The corresponding stability and bifurcation conditions of the period motions are predicted through eigenvalue analysis. Numerical simulation of the period motions in the double pendulum is completed from analytical predictions.

Analytical Prediction of Period-1 Motions in a Time-Delayed, Softening Duffing Oscillator

2017 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyuan Xing
Keyword(s):  
Differential Equation ◽  
Numerical Simulation ◽  
Duffing Oscillator ◽  
Mapping Method ◽  
Eigenvalue Analysis ◽  
Analytical Prediction ◽  
Implicit Mapping ◽  
Stability And Bifurcations ◽  
The Stability

In this paper, symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, softening Duffing oscillator is analytically predicted through a discrete implicit mapping method. Such a method is based on the discretization of the corresponding differential equation. The stability and bifurcations of the symmetric and asymmetric period-1 motions are determined through eigenvalue analysis. Numerical simulation of the period-1 motions in the time-delayed softening Duffing oscillator is presented for verification of the analytical prediction.

Independent Asymmetric Period-3 Motions in a Periodically Forced Double-Pendulum

2019 ◽  
Author(s):  
Albert C. J. Luo ◽  
Chuan Guo
Keyword(s):  
Numerical Simulation ◽  
Vibration Reduction ◽  
Mapping Method ◽  
Eigenvalue Analysis ◽  
Double Pendulum ◽  
Pendulum System ◽  
Implicit Mapping ◽  
Stability And Bifurcation

Abstract In this paper, the independent asymmetric period-3 motions of a periodically forced, damped, double-pendulum are predicted through a discrete implicit mapping method. The corresponding stability and bifurcation conditions of the paired asymmetric period-3 motions are determined through eigenvalue analysis. Numerical simulation of the two asymmetric period-3 motions in the double-pendulum system is completed from analytical predictions. The example presented herein can be used for the vibration reduction of the first pendulum through the motions of the second pendulum.

scholarly journals Statistical problem of ideal gas in general two-dimensional regions

2014 ◽  
Vol 29 (02) ◽  
pp. 1450243 ◽  
Author(s):  
Ci Song ◽  
Wen-Du Li ◽  
Pardon Mwansa ◽  
Ping Zhang
Keyword(s):  
Perturbation Theory ◽  
Conformal Mapping ◽  
Ideal Gas ◽  
Mapping Method ◽  
Numerical Results ◽  
Statistical Problem ◽  
Thermodynamic Quantities ◽  
Two Dimensional ◽  
Conformal Mapping Method

In this paper, based on the conformal mapping method and the perturbation theory, we develop a method to solve the statistical problem within general two-dimensional regions. We consider some examples and the numerical results and fitting results are given. We also give the thermodynamic quantities of the general two-dimensional regions, and compare the thermodynamic quantities of the different regions.

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.

Instability of unsteady flows or configurations. Part 2. Convective instability

1972 ◽  
Vol 54 (1) ◽  
pp. 143-152 ◽  
Author(s):  
Chia-Shun Yih ◽  
Chin-Hsiu Li
Keyword(s):  
Prandtl Number ◽  
Convective Instability ◽  
Floquet Theory ◽  
Critical Rayleigh Number ◽  
Unsteady Flows ◽  
Temperature Oscillation ◽  
Numerical Results ◽  
Periodic Temperature ◽  
Convective Cells ◽  
Time Periodic

The formation of convective cells in a fluid between two horizontal rigid boundaries with time-periodic temperature distribution is studied by the use of the Floquet theory. Numerical results for the critical Rayleigh number are given for a Prandtl number of 0·73 (air) and for various values of the frequency and magnitude of the primary temperature oscillation. Some numerical results for a Prandtl number of 7·0 (water) are also given. The most striking feature of the results is that the disturbances (or convection cells) oscillate either synchronously or with half frequency.

Periodic Motions and Bifurcation Trees in a Parametric Duffing Oscillator

2017 ◽  
Author(s):  
Albert C. J. Luo ◽  
Haolin Ma
Keyword(s):  
Differential Equation ◽  
Numerical Simulations ◽  
Duffing Oscillator ◽  
Eigenvalue Analysis ◽  
Analytic Method ◽  
Periodic Motions ◽  
Implicit Mapping ◽  
Stability And Bifurcation

This paper studies bifurcation trees of periodic motions in a parametric, damped Duffing oscillator. From the semi-analytic method, the corresponding differential equation is discretized to obtain the implicit mapping. From implicit mapping structure, the periodic nodes of periodic motions are computed, and the bifurcation trees of period-1 to period-4 motions are presented and the corresponding stability and bifurcation are carried out by eigenvalue analysis. From the analytical predictions, numerical simulations are completed, and the trajectory, harmonic amplitudes and phases of period-1 to period-4 motions are illustrated.

Complete Bifurcation Trees of Independent Period-2 Motions to Chaos in a Parametrically Excited Pendulum

2019 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Bifurcation Analysis ◽  
Mapping Method ◽  
Eigenvalue Analysis ◽  
Periodic Motions ◽  
Parametrically Excited ◽  
Stability And Bifurcation ◽  
Period 2 ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

Abstract In this paper, bifurcation trees of independent period-2 motions to chaos are investigated in a parametrically excited pendulum. The implicit discrete mapping method is employed to obtain periodic motions in such a system. Analytical predictions of periodic motions are based on the mapping structures and peroidicity. The bifurcation trees of independent period-2 motions to chaos are studied, and the corresponding stability and bifurcation analysis are completed through eigenvalue analysis. Finally, sampled period-2 motions are simulated numerically in comparison to the analytical predictions. The infinite bifurcation trees of independent period-2 motions to chaos can be obtained.

Stable and unstable nonlinear resonant response of hanging chains: theory and experiment

1993 ◽  
Vol 440 (1909) ◽  
pp. 345-364 ◽  
Keyword(s):  
Excitation Frequency ◽  
Three Dimensional ◽  
Harmonic Excitation ◽  
Numerical Results ◽  
Experimental Results ◽  
Distinct Region ◽  
Asymptotic Results ◽  
Sensitive Dependence ◽  
Theoretical Predictions ◽  
Hanging Chain

The three-dimensional nonlinear dynamics of a hanging chain, driven by harmonic excitation at the top, are studied first analytically and numerically, and then experimentally. Asymptotic results demonstrate a sensitive dependence on excitation frequency and amplitude. For moderately large excitation amplitudes there are distinct regions of stable two-dimensional and stable three-dimensional response as function of frequency, as well as a distinct region in which all steady-state solutions are unstable. Numerical results were obtained to verify the asymptotic solutions and investigate the dynamics within the irregular response region. Numerical results for even larger excitation amplitudes showed that large impulse-like tension forces cause the chain to lose tension over a region adjacent to its freely hanging end, and then collapse. Following the collapse, the chain configuration intersects itself. Experimental results confirm qualitatively and quantitatively the theoretical predictions. The experimental results also demonstrate the loss of tension and subsequent collapse of the chain at the predicted excitation amplitudes, as well as the intersection of the chain with itself.

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