Nonlinear Phase-Based Oscillator to Generate and Assist Periodic Motions

2016 ◽  
Author(s):  
Juan De la Fuente ◽  
Thomas G. Sugar ◽  
Sangram Redkar ◽  
Andrew R. Bates
Keyword(s):  
Phase Angle ◽  
Nonlinear Oscillator ◽  
Mechanical Systems ◽  
Oscillatory Behavior ◽  
Periodic Motions ◽  
Nonlinear Phase ◽  
Forcing Function ◽  
Hip Motion ◽  
The Stability ◽  
Bendixson Criterion

Oscillatory behavior is important for tasks such as walking and running. We are developing methods to add energy to enhance or vary the oscillatory behavior based on the system’s phase angle. We define a nonlinear oscillator using a forcing function based on the sine and cosine of the system’s phase angle that can modulate the amplitude and frequency of oscillation. The stability of the system is proved using the Poincaré-Bendixson criterion. Linear and rotational mechanical systems are simulated using our phase controller. The method is implemented and tested to control a pendulum. Lastly, we propose how to assist hip motion during walking using the phase-based forcing function.

Nonlinear, Phase-Based Oscillator to Generate and Assist Periodic Motions

2017 ◽  
Vol 9 (2) ◽  
Author(s):  
Juan De la Fuente ◽  
Thomas G. Sugar ◽  
Sangram Redkar
Keyword(s):  
Limit Cycle ◽  
Phase Angle ◽  
Trajectory Planning ◽  
Nonlinear Oscillator ◽  
Oscillatory Behavior ◽  
Periodic Motions ◽  
Nonlinear Phase ◽  
Forcing Function ◽  
Line Trajectory ◽  
Bendixson Criterion

Oscillatory behavior is important for tasks, such as walking and running. We are developing methods for wearable robotics to add energy to enhance or vary the oscillatory behavior based on the system's phase angle. We define a nonlinear oscillator using a forcing function based on the sine and cosine of the system's phase angle that can modulate the amplitude and frequency of oscillation. This method is based on the state of the system and does not use off-line trajectory planning. The behavior of a limit cycle is shown using the Poincaré–Bendixson criterion. Linear and rotational models are simulated using our phase controller. The method is implemented and tested to control a pendulum.

Period-1 Motions in a Two-Degree-of-Freedom Nonlinear Oscillator With Periodic Excitation

2015 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Analytical Solutions ◽  
Nonlinear Oscillator ◽  
Degrees Of Freedom ◽  
Harmonic Amplitude ◽  
Periodic Excitation ◽  
Periodic Motions ◽  
Two Degrees Of Freedom ◽  
Approximate Analytical Solutions ◽  
The Stability ◽  
Two Degree Of Freedom

In this paper, analytical solutions for period-1 motions in a periodically forced, two-degrees-of-freedom system with a nonlinear spring are developed. The stability and bifurcation of the periodic motions are completed by the eigenvalue analysis. Both symmetric and asymmetric periodic motions are found in the system. Analytical solutions of both stable and unstable period-1 are presented. Finally, numerical simulations of stable and unstable motions in the two degrees of freedom systems are presented. The harmonic amplitude spectrums show the harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be observed.

Analytical Period-1 Motions in a Periodically Forced Oscillator With Quadratic Nonlinearity

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Analytical Solutions ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Harmonic Balance Method ◽  
Series Expansions ◽  
Periodic Motions ◽  
Approximate Analytical Solutions ◽  
Forced Oscillator ◽  
The Stability ◽  
Generalized Harmonic Balance Method

In this paper, period-1 motions in a quadratic nonlinear oscillator under excitation are investigated by the generalized harmonic balance method. The analytical solutions of period-1 motion for such an oscillator are presented by the Fourier series expansions. The stability and bifurcation analysis of period-1 motion is carried out via eigenvalue analysis. To verify the approximate analytical solutions, numerical simulations are performed for a better understanding of the parameter characteristics of the period-1 solutions, and the stable and unstable periodic motions are illustrated. The analytical period-1 solutions are different from the perturbation analysis.

Investigation of the stability of periodic motions in a nonlinear automatic control system based on the fast fourier transform

2003 ◽  
Vol 3 ◽  
pp. 297-307
Author(s):  
V.V. Denisov
Keyword(s):  
Fourier Transform ◽  
Control System ◽  
Automatic Control ◽  
Fast Fourier Transform ◽  
Automatic Control System ◽  
Resonance Phenomenon ◽  
Periodic Motions ◽  
Multiply Connected ◽  
Non Linear ◽  
The Stability

An approach to the study of the stability of non-linear multiply connected systems of automatic control by means of a fast Fourier transform and the resonance phenomenon is considered.

Experimentally Determined Stability Parameters of a Subsonic Cascade Oscillating Near Stall

1978 ◽  
Vol 100 (1) ◽  
pp. 111-120 ◽  
Author(s):  
F. O. Carta ◽  
A. O. St. Hilaire
Keyword(s):  
Phase Angle ◽  
Incidence Angle ◽  
Minor Effect ◽  
Significant Finding ◽  
Force Coefficient ◽  
Stability Parameters ◽  
Free System ◽  
Pressure Transducers ◽  
Moment Coefficient ◽  
The Stability

Tests were performed on a linear cascade of airfoils oscillating in pitch about their midchords at frequencies up to 17 cps, at free-stream velocities up to 200 ft/s, and at interblade phase angles of 0 deg and 45 deg, under conditions of high aerodynamic loading. The measured data included unsteady time histories from chordwise pressure transducers and from chordwise hot films. Unsteady normal force coefficient, moment coefficient, and aerodynamic work per cycle of oscillation were obtained from integrals of the pressure data, and indications of the nature and extent of the separation phenomenon were obtained from an analysis of the hot-film response data. The most significant finding of this investigation is that a change in interblade phase angle from 0 deg to 45 deg radically alters the character of the unsteady blade loading (which governs its motion in a free system) from stable to unstable. Furthermore, the stability or instability is governed primarily by the phase angle of the pressure distribution (relative to the blade motion) over the forward 10–15 percent of the blade chord. Reduced frequency and mean incidence angle changes were found to have a relatively minor effect on stability for the range of parameters tested.

Bifurcation Trees of Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator

2014 ◽  
Vol 24 (05) ◽  
pp. 1450075 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Fourier Series ◽  
Nonlinear Systems ◽  
Analytical Solutions ◽  
Nonlinear Oscillator ◽  
Series Solution ◽  
Parametric Oscillator ◽  
Periodic Motions ◽  
Parametric Oscillators ◽  
Harmonic Term ◽  
Nonlinear Behaviors

In this paper, bifurcation trees of periodic motions to chaos in a parametric oscillator with quadratic nonlinearity are investigated analytically as one of the simplest parametric oscillators. The analytical solutions of periodic motions in such a parametric oscillator are determined through the finite Fourier series, and the corresponding stability and bifurcation analyses for periodic motions are completed. Nonlinear behaviors of such periodic motions are characterized through frequency–amplitude curves of each harmonic term in the finite Fourier series solution. From bifurcation analysis of the analytical solutions, the bifurcation trees of periodic motion to chaos are obtained analytically, and numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum. This investigation shows period-1 motions exist in parametric nonlinear systems and the corresponding bifurcation trees to chaos exist as well.

Galloping Vibration of Nonlinear Cables Through a Two Degree-of-Freedom Oscillator

2016 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Transmission Line ◽  
Analytical Solutions ◽  
Transmission Lines ◽  
Nonlinear Oscillator ◽  
Numerical Solutions ◽  
Degree Of Freedom ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Two Degree Of Freedom

In this paper, galloping vibrations of a lightly iced transmission line are investigated through a two-degree-of-freedom (2-DOF) nonlinear oscillator. The 2-DOF nonlinear oscillator is used to describe the transverse and torsional motions of the galloping cables. The analytical solutions of periodic motions of galloping cables are presented through generalized harmonic balanced method. The analytical solutions of periodic motions for the galloping cable are compared with the numerical solutions, and the corresponding stability and bifurcation of periodic motions are analyzed by the eigenvalues analysis. To demonstrate the accuracy of the analytical solutions of periodic motions, the harmonic amplitudes are presented. This investigation will help one better understand galloping mechanism of iced transmission lines.

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.

scholarly journals Unsteady Gapwise Periodicity of Oscillating Cascaded Airfoils

1982 ◽  
Author(s):  
F. O. Carta
Keyword(s):  
Unsteady Flow ◽  
Fourier Coefficient ◽  
Phase Angle ◽  
Leading Edge ◽  
Experimental Measurements ◽  
Pressure Data ◽  
Linear Cascade ◽  
Aerodynamic Damping ◽  
The Stability ◽  
Interblade Phase Angle

Tests were conducted on a linear cascade of airfoils oscillating in pitch to measure the unsteady pressure response on selected blades along the leading edge plane of the cascade and over the chord of the center blade. The pressure data were reduced to Fourier coefficient form for direct comparison, and were also processed to yield integrated loads and, particularly, the aerodynamic damping coefficient. In addition, results from two unsteady theories for cascaded blades with nonzero thickness and camber were compared with the experimental measurements. The three primary results that emerged from this investigation were: (a) from the leading edge plane blade data, the cascade was judged to be periodic in unsteady flow over the range of parameters tested, (b) as before, the interblade phase angle was found to be the single most important parameter affecting the stability of the oscillating cascade blades, and (c) the real blade theory and the experiment were in excellent agreement for the several cases chosen for comparison.

Dynamics of Two Van Der Pol Oscillators Coupled via a Bath

2003 ◽  
Author(s):  
Erika Camacho ◽  
Richard Rand ◽  
Howard Howland
Keyword(s):  
Software Package ◽  
Bifurcation Theory ◽  
Floquet Theory ◽  
Van Der Pol Oscillator ◽  
Direct Connection ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Existence And Stability ◽  
Van Der Pol Oscillators ◽  
The Stability

In this work we study a system of two van der Pol oscillators, x and y, coupled via a “bath” z: x¨−ε(1−x2)x˙+x=k(z−x)y¨−ε(1−y2)y˙+y=k(z−y)z˙=k(x−z)+k(y−z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

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