Dynamics of a Duffing Oscillator With Two Time Delays in Feedback Control Under Narrow-Band Random Excitation

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Yanfei Jin ◽  
Haiyan Hu
Keyword(s):  
Feedback Control ◽  
Time Delays ◽  
Multiple Scales ◽  
Narrow Band ◽  
Duffing Oscillator ◽  
Primary Resonance ◽  
Random Excitation ◽  
Numerical Results ◽  
Linear Feedback ◽  
The Difference

The paper presents analytical and numerical results of the primary resonance of a Duffing oscillator with two distinct time delays in the linear feedback control under narrow-band random excitation. Using the method of multiple scales, the first-order and the second-order steady-state moments of the primary resonance are derived. For the case of two distinct time delays, the appropriate choices of the combinations of the feedback gains and the difference between two time delays are discussed from the viewpoint of vibration control and stability. The analytical results are in well agreement with the numerical results.

Primary Resonance of a Duffing Oscillator With Two Distinct Time Delays in State Feedback Under Narrow-Band Random Excitation

2007 ◽  
Author(s):  
Yanfei Jin ◽  
Haiyan Hu
Keyword(s):  
Time Delays ◽  
State Feedback ◽  
Multiple Scales ◽  
Narrow Band ◽  
Damping Ratio ◽  
Duffing Oscillator ◽  
Primary Resonance ◽  
Random Excitation ◽  
Equivalent Damping ◽  
The Difference

The primary resonance of a Duffing oscillator with two distinct time delays in state feedback under narrow-band random excitation is investigated in detail by using the method of multiple scales. First, the equations of modulation of response amplitude and phase are determined. Then, the expressions of the first-order and the second-order steady-state moments and their stable regions are obtained by introducing the equivalent detuning frequency and the equivalent damping ratio. For the case of two distinct time delays, the appropriate choices of the combinations of the feedback gains and the difference between two time delays are discussed from the viewpoint of vibration control. Finally, the theoretical analyses are well verified through numerical simulations.

scholarly journals Optical continuum photometric reverberation mapping of the Seyfert-1 galaxy Mrk509

2019 ◽  
Author(s):  
F Pozo Nuñez ◽  
N Gianniotis ◽  
J Blex ◽  
T Lisow ◽  
R Chini ◽  
...  
Keyword(s):  
Time Delays ◽  
Narrow Band ◽  
Large Field ◽  
Black Hole Mass ◽  
Disk Model ◽  
Reverberation Mapping ◽  
Line Region ◽  
Optical Continuum ◽  
The Difference ◽  
Flux Calibration

Abstract We present the results of a two year optical continuum photometric reverberation mapping campaign carried out on the nucleus of the Seyfert-1 galaxy Mrk509. Specially designed narrow-band filters were used in order to mitigate the line and pseudo-continuum contamination of the signal from the broad line region, while allowing for high-accuracy flux-calibration over a large field of view. We obtained light curves with a sub-day time sampling and typical flux uncertainties of 1%. The high photometric precision allowed us to measure inter-band continuum time delays of up to ∼2 days across the optical range. The time delays are consistent with the relation τ∝λ4/3 predicted for an optically thick and geometrically thin accretion disk model. The size of the disk is, however, a factor of 1.8 larger than predictions based on the standard thin-disk theory. We argue that, for the particular case of Mrk509, a larger black hole mass due to the unknown geometry scaling factor can reconcile the difference between the observations and theory.

Resonances of a Nonlinear Single-Degree-of-Freedom System with Time Delay in Linear Feedback Control

2010 ◽  
Vol 65 (5) ◽  
pp. 357-368 ◽  
Author(s):  
Atef F. El-Bassiouny ◽  
Salah El-Kholy
Keyword(s):  
Steady State ◽  
Time Delay ◽  
Feedback Control ◽  
Fixed Points ◽  
Multiple Scales ◽  
Degree Of Freedom ◽  
Linear Feedback ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Subharmonic Resonances

The primary and subharmonic resonances of a nonlinear single-degree-of-freedom system under feedback control with a time delay are studied by means of an asymptotic perturbation technique. Both external (forcing) and parametric excitations are included. By means of the averaging method and multiple scales method, two slow-flow equations for the amplitude and phase of the primary and subharmonic resonances and all other parameters are obtained. The steady state (fixed points) corresponding to a periodic motion of the starting system is investigated and frequency-response curves are shown. The stability of the fixed points is examined using the variational method. The effect of the feedback gains, the time-delay, the coefficient of cubic term, and the coefficients of external and parametric excitations on the steady-state responses are investigated and the results are presented as plots of the steady-state response amplitude versus the detuning parameter. The results obtained by two methods are in excellent agreement

Energy Harvesting From Vibrations With a Nonlinear Oscillator

2010 ◽  
Vol 132 (2) ◽  
Author(s):  
David A. W. Barton ◽  
Stephen G. Burrow ◽  
Lindsay R. Clare
Keyword(s):  
Energy Harvesting ◽  
Pure Tone ◽  
Nonlinear Oscillator ◽  
Narrow Band ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Type Equation ◽  
Random Excitation ◽  
Resonant Response ◽  
Electromagnetic Energy Harvesting

In this paper, we present a nonlinear electromagnetic energy harvesting device that has a broadly resonant response. The nonlinearity is generated by a particular arrangement of magnets in conjunction with an iron-cored stator. We show the resonant response of the system to both pure-tone excitation and narrow-band random excitation. In addition to the primary resonance, the superharmonic resonances of the harvester are also investigated and we show that the corresponding mechanical upconversion of the excitation frequency may be useful for energy harvesting. The harvester is modeled using a Duffing-type equation and the results are compared with the experimental data.

scholarly journals Hybrid rayleigh–van der pol–duffing oscillator: Stability analysis and controller

2021 ◽  
pp. 146134842110264
Author(s):  
Chun-Hui He ◽  
Dan Tian ◽  
Galal M Moatimid ◽  
Hala F Salman ◽  
Marwa H Zekry
Keyword(s):  
Fixed Points ◽  
Multiple Scales ◽  
Duffing Oscillator ◽  
Primary Resonance ◽  
Time History ◽  
Phase Portraits ◽  
Response Curves ◽  
Van Der Pol ◽  
Stability Performance ◽  
Linearized Stability

The current study examines the hybrid Rayleigh–Van der Pol–Duffing oscillator (HRVD) with a cubic–quintic nonlinear term and an external excited force. The Poincaré–Lindstedt technique is adapted to attain an approximate bounded solution. A comparison between the approximate solution with the fourth-order Runge–Kutta method (RK4) shows a good matching. In case of the autonomous system, the linearized stability approach is employed to realize the stability performance near fixed points. The phase portraits are plotted to visualize the behavior of HRVD around their fixed points. The multiple scales method, along with a nonlinear integrated positive position feedback (NIPPF) controller, is employed to minimize the vibrations of the excited force. Optimal conditions of the operation system and frequency response curves (FRCs) are discussed at different values of the controller and the system parameters. The system is scrutinized numerically and graphically before and after providing the controller at the primary resonance case. The MATLAB program is employed to simulate the effectiveness of different parameters and the controller on the system. The calculations showed that NIPPF is the best controller. The validations of time history and FRC of the analysis as well as the numerical results are satisfied by making a comparison among them.

Bifurcation of a Duffing Oscillator Having Nonlinear Fractional Derivative Feedback

2014 ◽  
Vol 24 (03) ◽  
pp. 1450028 ◽  
Author(s):  
A. Y. T. Leung ◽  
H. X. Yang ◽  
P. Zhu
Keyword(s):  
Feedback Control ◽  
Control Strategies ◽  
Duffing Oscillator ◽  
Primary Resonance ◽  
Nonlinear Feedback ◽  
Flow Equations ◽  
First Order ◽  
Active Feedback ◽  
Active Feedback Control ◽  
Subharmonic Resonances

Active feedback control is commonly used to attenuate undesired vibrations in vibrating machineries and structures, such as bridges, highways and aircrafts. In this paper, we investigate the primary resonance and 1/3 subharmonic resonance of a harmonically forced Duffing oscillator under fractional nonlinear feedback control. By means of the first order averaging method, slow flow equations governing the modulations of amplitude and phase of the oscillator are obtained. An approximate solution for the steady state periodic response is derived and its stability is determined by the Routh–Hurwitz criterion. We demonstrate that appropriate choices on the control strategies and feedback gains can delay or eliminate the undesired bifurcations and reduce the amplitude peak both of the primary and subharmonic resonances. Analytical results are verified by comparisons with the numerical integration results.

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