scholarly journals Stability and bifurcation analysis of super- and sub-harmonic responses in a torsional system with piecewise-type nonlinearities

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Jong-Yun Yoon ◽  
Byeongil Kim
Keyword(s):  
Nonlinear Dynamic ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Stability Conditions ◽  
Arc Length ◽  
Dynamic Behaviors ◽  
Variational Stability ◽  
Time Histories ◽  
The Stability

AbstractThe nonlinear dynamic behaviors induced by piecewise-type nonlinearities generally reflect super- and sub-harmonic responses. Various inferences can be drawn from the stability conditions observed in nonlinear dynamic behaviors, especially when they are projected in physical motions. This study aimed to investigate nonlinear dynamic characteristics with respect to variational stability conditions. To this end, the harmonic balance method was first implemented by employing Hill’s method, and the time histories under stable and unstable conditions were examined using a numerical simulation. Second, the super- and sub-harmonic responses were investigated according to frequency upsweeping based on the arc-length continuation method. While the stability conditions vary along the arc length, the bifurcation phenomena also show various characteristics depending on their stable or unstable status. Thus, the study findings indicate that, to determine the various stability conditions along the direction of the arc length, it is fairly reasonable to determine nonlinear dynamic behaviors such as period-doubling, period-doubling cascade, and quasi-periodic (or chaotic) responses. Overall, this study suggests analytical and numerical methods to understand the super- and sub-harmonic responses by comparing the arc length of solutions with the variational stability conditions.

Nonlinear Vibration Analysis of Viscoelastic Isolator

2012 ◽  
Vol 160 ◽  
pp. 140-144
Author(s):  
Chao Zhou ◽  
Cai Mao Zhong
Keyword(s):  
Dynamic Response ◽  
Fractional Derivative ◽  
Nonlinear Dynamic ◽  
Harmonic Balance Method ◽  
Nonlinear Damping ◽  
Vibration Isolator ◽  
Response Characteristics ◽  
Nonlinear Dynamic Equation ◽  
The Stability ◽  
Stiffness And Damping

Research on nonlinear dynamic response of passive vibration isolator, which was excited by foundation vibration and isolated by viscoelastic material was done. Nonlinear stiffness was expressed by the cubic polynomial function of deformation and nonlinear damping was characterized by viscoelastic fractional derivative operator. Then the fractional derivative nonlinear dynamic equation of passive vibration isolator was established. The dynamic response characteristics were analyzed by harmonic balance method and the frequency response equation and amplitude-frequency curve were obtained, and furthermore, the influence of nonlinearity on system was analyzed. Finally, the stability and the stable interval of the periodic solution were argued by the Floquet theory. The result s indicates that the proposed equation can precisely describe the dynamic characteristics of viscoelastic vibration isolator. The ignorance of nonlinearity of stiffness and damping will result in obvious error. The proposed method provides theoretic reference for design of viscoelastic isolator and the evaluation of its effect.

ON THE NUMERICAL DETECTION OF CODIMENSION-3 BIFURCATIONS OF PERIODIC SOLUTIONS (WITH APPLICATION TO OPTICAL BISTABILITY)

1994 ◽  
Vol 04 (06) ◽  
pp. 1425-1446
Author(s):  
KLAUS-GEORG NOLTE ◽  
IVAN L’HEUREUX
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Optical Bistability ◽  
Continuation Method ◽  
Period Doubling ◽  
Bistable System ◽  
Dimensional System ◽  
Systems Of Equations ◽  
The Stability ◽  
Stationary Behavior

Based upon the combination of the pseudo-arclength continuation method and the Poincaré map defined on a variable return plane, systems of equations are constructed that trace a Takens-Bogdanov bifurcation, a cusp, an isola formation/perturbed bifurcation point and a degenerate period-doubling/secondary Hopf bifurcation of periodic solutions of autonomous ordinary differential equations. The implementation of these ideas into a collection of FORTRAN codes and its application to a five-dimensional system describing an optical bistable system lead to the detection of interesting codimension-3 bifurcations away from the stationary behavior. A winged cusp, a swallow tail, a degenerate hysteresis point, an isola formation point for a codimension-1 loop and two kinds of degenerate Takens-Bogdanov bifurcations of periodic solutions are presented. Finally, based upon the computation of the stability coefficient “a”, attractive tori are found in a systematic way and briefly discussed.

Dynamic Simulation of a Woodpecker Robot Based on Self-Excited Vibration

2013 ◽  
Vol 753-755 ◽  
pp. 2020-2024
Author(s):  
Yong Xu ◽  
Wei Hu ◽  
Shan Ping Zhang
Keyword(s):  
Dynamic Simulation ◽  
Nonlinear Dynamic ◽  
Passive Movement ◽  
Robot System ◽  
Dynamic Simulations ◽  
Dynamic Behaviors ◽  
Excited Vibration ◽  
Time Histories ◽  
Stable Passive

This paper studies complex nonlinear dynamic behaviors of a woodpecker robot system which can only operate in the presence of friction as it relies on combined impacts and jamming. The woodpecker robot can periodically move without any drives and controls based on self-excited vibration phenomena. The whole time histories of the dynamic simulations in successive periods indicate its cyclical, stable passive movement. Keywords: Passive movement; Self-excited vibration; Dynamic simulation; impact and friction

Chaotic Vibration of Fractional Calculus Model Viscoelastic Arch

2014 ◽  
Vol 668-669 ◽  
pp. 151-155
Author(s):  
Fan Huang ◽  
Jun Huang ◽  
Shao Bin Dai
Keyword(s):  
Fractional Calculus ◽  
Forced Vibration ◽  
Periodic Motion ◽  
Nonlinear Dynamic ◽  
Material Parameter ◽  
Load Parameter ◽  
Motion Equations ◽  
Dynamic Behaviors ◽  
Chaotic Vibration ◽  
The Stability

Nonlinear dynamic behaviors of the fractional calculus model viscoelastic arch are discussed in this paper. The motion equations governing dynamic behaviors of the viscoelastic arch is derived and simplified by Galerkin method. The numerical method for solving the motion equations with fractional calculus is presented here. The influences of the load parameter and the material parameter are considered respectively. The results show that the chaotic vibration of the arch appears in forced vibration. And both the load parameter and the material parameter affect the dynamic behavior of the arch. With the increasing of the load parameter, the motion states changed from periodic motion with period 1 to complex motions, such as mult-periodicity, quasi-periodicity or chaos. The increasing of the material parameter benefits the stability of the structures.

Step-length control analysis with a multi-staged clutch damper model in a torsional vibration problem using harmonic balance method

2015 ◽  
Vol 230 (12) ◽  
pp. 1968-1977
Author(s):  
Jong-Yun Yoon ◽  
Byeongil Kim
Keyword(s):  
Nonlinear Dynamic ◽  
Control Method ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Nonlinear Dynamic System ◽  
Step Length ◽  
Dynamic Responses ◽  
Clear Understanding ◽  
Control Analysis ◽  
Adaptive Step

This study focuses on a step-length control method to investigate piecewise type nonlinearities such as multi-staged clutch dampers in a practical system. In general, step-length control techniques are employed to overcome at least two difficulties with respect to examining the nonlinear dynamic responses of a physical system. Such techniques first reduce the calculation time, which increases the efficiency of simulation in many case studies. Then, the employment of step-length control resolves strongly stiff problems which often occur in the asymmetrical piecewise type nonlinearities. In order to overcome these difficulties, this study suggests a newly developed step-length control method based on the employment of two different adaptive step sizes from prior steps to the next steps, when simulation is conducted using an arc-length continuation method. The simulation results using the step-length control method show much improvement with respect to reducing the calculation times and number of steps compared with the ones without step-length control method. Thus, this study provides clear understanding of the objectives of using step-length control such that it increases the efficiency of simulation as well as resolves the convergence problems in a nonlinear dynamic system.

Nonlinear Vibrations of Buried Rectangular Plate

2018 ◽  
Vol 140 (5) ◽  
Author(s):  
Guangyang Hong ◽  
Jian Li ◽  
Zhicong Luo ◽  
Hongying Li
Keyword(s):  
Nonlinear Dynamic ◽  
Granular Layer ◽  
Excitation Frequency ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Dynamic Strain ◽  
Burial Depth ◽  
Dynamic Behaviors ◽  
Jump Phenomena ◽  
Nonlinear Behaviors

We perform an investigation on the vibration response of a simply supported plate buried in glass particles, focusing on the nonlinear dynamic behaviors of the plate. Various excitation strategies, including constant-amplitude variable-frequency sweep and constant-frequency variable-amplitude sweep are used during the testing process. We employ the analysis methods of power spectroscopy, phase diagramming, and Poincare mapping, which reveal many complicated nonlinear behaviors in the dynamic strain responses of an elastic plate in granular media, such as the jump phenomena, period-doubling bifurcation, and chaos. The results indicate that the added mass, damping, and stiffness effects of the granular medium on the plate are the source of the nonlinear dynamic behaviors in the oscillating plate. These nonlinear behaviors are related to the burial depth of the plate (the thickness of the granular layer above plate), force amplitude, and particle size. Smaller particles and a suitable burial depth cause more obvious jump and period-doubling bifurcation phenomena to occur. Jump phenomena show both soft and hard properties near various resonant frequencies. With an increase in the excitation frequency, the nonlinear added stiffness effect of the granular layer makes a transition from strong negative stiffness to weak positive stiffness.

Bifurcation Analysis of a Belousov–Zhabotinsky Reaction Model

2015 ◽  
Vol 25 (06) ◽  
pp. 1550093 ◽  
Author(s):  
Xiaoli Wang ◽  
Yu Chang ◽  
Dashun Xu
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Reaction Model ◽  
Bifurcation Phenomena ◽  
Route To Chaos ◽  
The Stability ◽  
Belousov Zhabotinsky Reaction

We investigate the bifurcation phenomena in a Belousov–Zhabotinsky reaction model by applying Hopf bifurcation theory in frequency domain and harmonic balance method. The high accurate predictions, i.e. fourth-order harmonic balance approximation, on frequencies, amplitudes, and approximation expressions for periodic solutions emerging from Hopf bifurcation are provided. We also detect the stability and location of these periodic solutions. Numerical simulations not only confirm the theoretical analysis results but also illustrate some complex oscillations such as a cascade of period-doubling bifurcation, quasi-periodic solution, and period-doubling route to chaos. All these results improve the understanding of the dynamics of the model.

Dynamic vibratory motion analysis of a multi-degree-of-freedom torsional system with strongly stiff nonlinearities

2014 ◽  
Vol 229 (8) ◽  
pp. 1399-1414 ◽  
Author(s):  
Jong-yun Yoon ◽  
Hyeongill Lee
Keyword(s):  
Numerical Simulation ◽  
Nonlinear Dynamic ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Degree Of Freedom ◽  
Balance Method ◽  
Nonlinear Frequency ◽  
Dynamic Behaviors ◽  
Resonance Frequencies ◽  
Torsional System

Physical driveline systems have inherent nonlinearities such as multiple piecewise linear springs, gear backlashes, and drag torques. The multi-staged clutch dampers, in particular, cause severe problems in simulating the nonlinear dynamic behaviors of multi-degree-of-freedom systems. In order to analyze the nonlinear dynamic behaviors of the system, the harmonic balance method has been employed. This study suggests a method to overcome the convergence problems with strong nonlinearities by employing two distinct smoothening factors for stiffness and hysteresis. First, the dynamic behaviors of the multi-degree-of-freedom torsional system are investigated by employing multi-staged clutch dampers subjected to a sinusoidal excitation. Second, the effects of system parameters are examined with respect to dynamic characteristics of torsional vibration. The regimes of resonance frequencies along with the relevant parameters of the system are investigated by calculating backbone curves, which reduce the calculation time significantly. In order to validate harmonic balance method simulation, the simulated results are compared with those of numerical simulation. Harmonic balance method is shown to be more efficient than numerical simulation in calculating the nonlinear frequency response, as well as in simulating the steady-state responses without transient response effect.

Passive Movement Modeling of a Woodpecker Robot

2013 ◽  
Vol 415 ◽  
pp. 23-25
Author(s):  
Kai Yan Zhang ◽  
Yong Xu
Keyword(s):  
Nonlinear Dynamic ◽  
Dynamic Parameter ◽  
Passive Movement ◽  
Robot System ◽  
Dynamic Behaviors ◽  
Excited Vibration ◽  
Time Histories ◽  
Stable Passive

This paper studies complex nonlinear dynamic behaviors of a woodpecker robot system which can only operate in the presence of friction as it relies on combined impacts and jamming. The woodpecker robot can periodically move without any drives and controls based on self-excited vibration principle. The whole time histories of the dynamic parameter simulations in successive periods indicate that the robot is able to achieve cyclical, stable passive movement.

scholarly journals Geometrical Non-Linear Periodic Vibration of Plates

2000 ◽  
Author(s):  
M. Petyt ◽  
P. Ribeiro
Keyword(s):  
Steady State ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Internal Resonances ◽  
Resonance Frequencies ◽  
Modal Coupling ◽  
Non Linear ◽  
The Stability

Abstract Periodic, geometrically non-linear free and steady-state forced vibrations of fully clamped plates are investigated. The hierarchical finite element method (HFEM) and the harmonic balance method are used to derive the equations of motion in the frequency domain, which are solved by a continuation method. It is demonstrated that the HFEM requires far fewer degrees of freedom than the h-version of the FEM. Internal resonances due to modal coupling between modes with resonance frequencies related by a rational number, are discovered. In free vibration, internal resonances cause a very significant variation of the mode shape during the period of vibration. A similar behaviour is observed in steady-state forced vibration. The stability of the steady-state solutions is studied by Floquet’s theory and it is shown that stable multi-modal solutions occur.

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