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.

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.

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.

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.

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.

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.

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.

Period-3 Motions in a Periodically Forced, Damped, Double-Well Duffing Oscillator With Time-Delay

2017 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyuan Xing
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Analytical Method ◽  
Duffing Oscillator ◽  
Eigenvalue Analysis ◽  
Algebraic Equations ◽  
Implicit Mapping ◽  
Mapping Structures ◽  
Nonlinear Algebraic Equations ◽  
Amplitude Characteristics

In this paper, period-3 motions in a double-well Duffing oscillator with time-delay are predicted by a semi-analytical method. The implicit mapping structures of period-3 motions are constructed through the implicit mappings obtained by discretization of the corresponding differential equation. Complex period-3 motions are predicted through nonlinear algebraic equations of the implicit mappings in the mapping structures and the corresponding stability and bifurcation are carried out through eigenvalue analysis. Numerical and analytical results of complex period-3 motions are obtained and the corresponding frequency-amplitude characteristics are presented.

scholarly journals The study of the numerical diffusion in computational calculation

2020 ◽  
Vol 310 ◽  
pp. 00039
Author(s):  
Kamila Kotrasova ◽  
Vladimira Michalcova
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Numerical Simulation ◽  
Continuity Equation ◽  
Cfd Simulation ◽  
Flow Process ◽  
Ansys Fluent ◽  
Numerical Diffusion ◽  
Mesh Density ◽  
Computational Calculation

The numerical simulation of flow process and heat transfer phenomena demands the solution of continuous differential equation and energy-conservation equations coupled with the continuity equation. The choosing of computation parameters in numerical simulation of computation domain have influence on accuracy of obtained results. The choose parameters, as mesh density, mesh type and computation procedures, for the numerical diffusion of computation domain were analysed and compared. The CFD simulation in ANSYS – Fluent was used for numerical simulation of 3D stational temperature flow of the computation domain.

Multi-Step Forward Finite Element Method for the Numerical Simulation of Sheet Metal Forming

2011 ◽  
Vol 474-476 ◽  
pp. 251-254
Author(s):  
Jian Jun Wu ◽  
Wei Liu ◽  
Yu Jing Zhao
Keyword(s):  
Numerical Simulation ◽  
Finite Element Method ◽  
Finite Element ◽  
Sheet Metal ◽  
Sheet Metal Forming ◽  
Deep Drawing ◽  
Metal Forming ◽  
Mapping Method ◽  
Constitutive Relationship ◽  
Element Method

The multi-step forward finite element method is presented for the numerical simulation of multi-step sheet metal forming. The traditional constitutive relationship is modified according to the multi-step forming processes, and double spreading plane based mapping method is used to obtain the initial solutions of the intermediate configurations. To verify the multi-step forward FEM, the two-step simulation of a stepped box deep-drawing part is carried out as it is in the experiment. The comparison with the results of the incremental FEM and test shows that the multi-step forward FEM is efficient for the numerical simulation of multi-step sheet metal forming processes.

SINGULARITY, SWITCHABILITY AND BIFURCATIONS IN A 2-DOF, PERIODICALLY FORCED, FRICTIONAL OSCILLATOR

2013 ◽  
Vol 23 (03) ◽  
pp. 1330009 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
MOZHDEH S. FARAJI MOSADMAN
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Degrees Of Freedom ◽  
Eigenvalue Analysis ◽  
Analytical Dynamics ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the analytical dynamics for singularity, switchability, and bifurcations of a 2-DOF friction-induced oscillator is investigated. The analytical conditions of the domain flow switchability at the boundaries and edges are developed from the theory of discontinuous dynamical systems, and the switchability conditions of boundary flows from domain and edge flows are presented. From the singularity and switchability of flow to the boundary, grazing, sliding and edge bifurcations are obtained. For a better understanding of the motion complexity of such a frictional oscillator, switching sets and mappings are introduced, and mapping structures for periodic motions are adopted. Using an eigenvalue analysis, the stability and bifurcation analysis of periodic motions in the friction-induced system is carried out. Analytical predictions and parameter maps of periodic motions are performed. Illustrations of periodic motions and the analytical conditions are completed. The analytical conditions and methodology can be applied to the multi-degrees-of-freedom frictional oscillators in the same fashion.

Sign in / Sign up

Export Citation Format

Share Document