Nonlinear Airfoil Limit Cycle Analysis Using Continuation Method and Filtered Impulse Function

AIAA Journal ◽  
2020 ◽  
Vol 58 (5) ◽  
pp. 1976-1991
Author(s):  
Q. Yu ◽  
M. Damodaran ◽  
B. C. Khoo
Keyword(s):  
Limit Cycle ◽  
Continuation Method ◽  
Impulse Function

scholarly journals On born of in-phase limit cycle in ensemble of excitatory coupled FitzHugh-Nagumo elements

2018 ◽  
Vol 20 (3) ◽  
pp. 273-281
Author(s):  
Alexander G. Korotkov ◽  
Tatiana A. Levanova
Keyword(s):  
Hopf Bifurcation ◽  
Phase Space ◽  
Limit Cycle ◽  
Phenomenological Model ◽  
Coupling Strength ◽  
Active Element ◽  
Smooth Approximation ◽  
Impulse Function

We proposed and studied numerically efficient phenomenological model of ensemble of two FitzHugh-Nagumo neuron-like elements that are coupled by symmetric synaptic excitatory coupling. This coupling is defined by function that depends on phase of active element and that is smooth approximation of rectangular impulse function. Above-mentioned coupling depends on three parameters that define the beginning of element activation, the duration of the activation and the coupling strength. We show analytically that in the phase space of the model there exists stable in-phase limit cycle that corresponds to regular oscillations with equal phases and frequencies of elements. It is proved that this limit cycle is a result of supercritical Andronov-Hopf bifurcation. The chart of activity regimes is depicted on the plane of parameters that define beginning and duration of activation. The boundaries of bifurcations that lead to birth of this cycle are found.

scholarly journals The Calculation Process of the Limit Cycle for Lorenz System

2021 ◽  
Author(s):  
Hao Dong ◽  
Bin Zhong
Keyword(s):  
Numerical Method ◽  
Limit Cycle ◽  
Equilibrium Solution ◽  
Lorenz System ◽  
Continuation Method ◽  
Equilibrium Solutions ◽  
Periodical Solution ◽  
Calculation Process ◽  
Continuation Algorithm ◽  
Bifurcation Behavior

This work focuses on the bifurcation behavior before chaos phenomenon happens. Traditional numerical method is unable to solve the unstable limit cycle of nonlinear system. One algorithm is introduced to solve the unstable one, which is based one of the continuation method is called DEPAR approach. Combined with analytic and numerical method, the two stable and symmetrical equilibrium solutions exist through Fork bifurcation and the unstable and symmetrical limit cycles exist through Hopf bifurcation of Lorenz system. With the continuation algorithm, the bifurcation behavior and its phase diagram is solved. The results demonstrate the unstable periodical solution is around the equilibrium solution, besides the trajectory into the unstable area cannot escape but only converge to the equilibrium solution.

scholarly journals On born of in-phase limit cycle in ensemble of excitatory coupled FitzHugh-Nagumo elements

2018 ◽  
Vol 20 (3) ◽  
pp. 273-281
Author(s):  
A. G. Korotkov ◽  
T. A. Levanova
Keyword(s):  
Hopf Bifurcation ◽  
Phase Space ◽  
Limit Cycle ◽  
Phenomenological Model ◽  
Coupling Strength ◽  
Active Element ◽  
Smooth Approximation ◽  
Impulse Function

We proposed and studied numerically efficient phenomenological model of ensemble of two FitzHugh-Nagumo neuron-like elements that are coupled by symmetric synaptic excitatory coupling. This coupling is defined by function that depends on phase of active element and that is smooth approximation of rectangular impulse function. Above-mentioned coupling depends on three parameters that define the beginning of element activation, the duration of the activation and the coupling strength. We show analytically that in the phase space of the model there exists stable in-phase limit cycle that corresponds to regular oscillations with equal phases and frequencies of elements. It is proved that this limit cycle is a result of supercritical Andronov-Hopf bifurcation. The chart of activity regimes is depicted on the plane of parameters that define beginning and duration of activation. The boundaries of bifurcations that lead to birth of this cycle are found.

Effects of nonlinear interactions of flexural modes on the performance of a beam autoparametric vibration absorber

2019 ◽  
Vol 26 (7-8) ◽  
pp. 459-474
Author(s):  
Saeed Mahmoudkhani ◽  
Hodjat Soleymani Meymand
Keyword(s):  
Frequency Response ◽  
Internal Resonance ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Vibration Absorber ◽  
Primary System ◽  
Lumped Mass ◽  
Response Curves ◽  
Internal Resonances ◽  
The One

The performance of the cantilever beam autoparametric vibration absorber with a lumped mass attached at an arbitrary point on the beam span is investigated. The absorber would have a distinct feature that in addition to the two-to-one internal resonance, the one-to-three and one-to-five internal resonances would also occur between flexural modes of the beam by tuning the mass and position of the lumped mass. Special attention is paid on studying the effect of these resonances on increasing the effectiveness and extending the range of excitation amplitudes at which the autoparametric vibration absorber remains effective. The problem is formulated based on the third-order nonlinear Euler–Bernoulli beam theory, where the assumed-mode method is used for deriving the discretized equations of motion. The numerical continuation method is then applied to obtain the frequency response curves and detect the bifurcation points. The harmonic balance method is also employed for detecting the type of internal resonances between flexural modes by inspecting the frequency response curves corresponding to different harmonics of the response. Parametric studies on the performance of the absorber are conducted by varying the position and mass of the lumped mass, while the frequency ratio of the primary system to the first mode of the beam is kept equal to two. Results indicated that the one-to-five internal resonance is especially responsible for the considerable enhancement of the performance.

Sign in / Sign up

Export Citation Format

Share Document