Non-smooth approximations of the limiting phase trajectories for the Duffing oscillator near 1:1 resonance

2011 ◽  
Vol 240 (1) ◽  
pp. 1-12 ◽  
Author(s):  
L.I. Manevitch ◽  
A.S. Kovaleva ◽  
D.S. Shepelev
Keyword(s):  
Duffing Oscillator ◽  
Phase Trajectories ◽  
Limiting Phase Trajectories ◽  
Smooth Approximations

scholarly journals Limiting Phase Trajectories and Resonance Energy Transfer in a System of Two Coupled Oscillators

2010 ◽  
Vol 2010 ◽  
pp. 1-24 ◽  
Author(s):  
L. I. Manevitch ◽  
A. S. Kovaleva ◽  
E. L. Manevitch
Keyword(s):  
Energy Transfer ◽  
Coupled Oscillators ◽  
Energy Exchange ◽  
Duffing Oscillator ◽  
Normal Modes ◽  
Resonance Energy ◽  
Harmonic Excitation ◽  
Conservation Of Energy ◽  
Phase Trajectories ◽  
Limiting Phase Trajectories

We study a problem of energy exchange in a system of two coupled oscillators subject to 1 : 1 resonance. Our results exploit the concept of limiting phase trajectories (LPTs). The LPT, associated with full energy transfer, is, in certain sense, an alternative to nonlinear normal modes characterized by conservation of energy. We consider two benchmark examples. As a first example, we construct an LPT and examine the convergence to stationary oscillations for a Duffing oscillator subjected to resonance harmonic excitation. As a second example, we treat resonance oscillations in a system of two nonlinearly coupled oscillators. We demonstrate the reduction of the equations of motion to an equation of a single oscillator. It is shown that the most intense energy exchange and beating arise when motion of the equivalent oscillator is close to an LPT. Damped beating and the convergence to rest in a system with dissipation are demonstrated.

Quantitative Analysis of Noise Impact on Duffing Chaotic Detection System Using Lyapunov Characteristic Exponents

2011 ◽  
Vol 130-134 ◽  
pp. 1331-1337
Author(s):  
Wen Jing Hu ◽  
Zhi Zhen Liu ◽  
Zhi Hui Li
Keyword(s):  
Detection System ◽  
Signal To Noise Ratio ◽  
Duffing Oscillator ◽  
Threshold Value ◽  
System State ◽  
Signal To Noise ◽  
Phase Trajectories ◽  
Lyapunov Characteristic Exponents ◽  
Noise Ratio ◽  
System States

Performance of the Duffing oscillator to detect weak signals buried in heavy noise is analyzed quantitatively by LCEs. First in the case of noise, differential equations to compute LCE s are derived using RHR algorithm, so the quantitative criteria to identify system states are obtained. Then using LCEs, the threshold value of the forced periodic term is found accurately. Finally the system state and state change are analyzed using LCEs by keeping the threshold value and varying the noise intensity, and the minimum signal to noise ratio is determined. By contrast of phase trajectories and LCEs, it shows that phase trajectories disturbed by strong noise sometimes are ambiguous to our eyes, but through LCEs, the system state can be identified clearly and quantitatively especially in strong noise background. So the minimum signal to noise ratio can be obtained accurately.

LIMITING PHASE TRAJECTORIES AND TRANSIENT RESONANCE OSCILLATIONS IN 1 AND 2 DOF ASYMMETRIC NONLINEAR SYSTEMS

2011 ◽  
Vol 21 (10) ◽  
pp. 2919-2928
Author(s):  
E. L. MANEVITCH ◽  
L. I. MANEVITCH
Keyword(s):  
Energy Exchange ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Conservation Of Energy ◽  
Phase Trajectories ◽  
Weakly Coupled ◽  
Limiting Phase Trajectories ◽  
Nonlinear Normal ◽  
Nonstationary Oscillations ◽  
Resonance Oscillations

The concept of limiting phase trajectories (LPT) has been introduced by one of the authors to describe intensive energy exchange between weakly coupled oscillators or oscillatory chains. It turns out that LPT can be considered as an alternative to nonlinear normal modes (NNMs), which are characterized by conservation of energy. LPT (in the introduced coordinates) describes the vibroimpact-type process with saw-tooth amplitude and a discontinuous derivative. It was shown earlier that this concept could also be extended to systems with one degree of freedom (DoF). In this case energy exchange between the oscillator and the source of energy can occur. In this paper, we generalize the above results in several ways, namely: (1) a consideration of the asymmetry of elastic potential; (2) a detailed description of the origin of vibroimpact-type behavior and the transition from nonresonant nonstationary oscillations to resonant ones (3) a direct application of obtained results to transient vibrations in strongly asymmetric 2DoF systems.

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