scholarly journals Phase and amplitude dynamics in large systems of coupled oscillators: Growth heterogeneity, nonlinear frequency shifts, and cluster states

2013 ◽  
Vol 23 (3) ◽  
pp. 033116 ◽  
Author(s):  
Wai Shing Lee ◽  
Edward Ott ◽  
Thomas M. Antonsen
Keyword(s):  
Coupled Oscillators ◽  
Nonlinear Frequency ◽  
Cluster States ◽  
Frequency Shifts ◽  
Large Systems

Stabilization of Explosive Instabilities by Nonlinear Frequency Shifts

1973 ◽  
Vol 30 (2) ◽  
pp. 49-51 ◽  
Author(s):  
V. N. Oraevskii ◽  
V. P. Pavlenko ◽  
H. Wilhelmsson ◽  
E. Ya. Kogan
Keyword(s):  
Nonlinear Frequency ◽  
Frequency Shifts

Linear and nonlinear frequency shifts in acoustical resonators with varying cross sections

2001 ◽  
Vol 110 (1) ◽  
pp. 109-119 ◽  
Author(s):  
Mark F. Hamilton ◽  
Yurii A. Ilinskii ◽  
Evgenia A. Zabolotskaya
Keyword(s):  
Cross Sections ◽  
Nonlinear Frequency ◽  
Frequency Shifts ◽  
Linear And Nonlinear

Effects of nonlinear frequency shifts on certain induced scattering processes

2002 ◽  
Vol 9 (11) ◽  
pp. 4520-4524 ◽  
Author(s):  
Peter H. Yoon ◽  
Rudi Gaelzer
Keyword(s):  
Nonlinear Frequency ◽  
Frequency Shifts

Effect of Negative Stiffness Content on the Periodic Motion of Nonlinearly Coupled Oscillators

2021 ◽  
Vol 16 (11) ◽  
Author(s):  
Mohammad A. Al-Shudeifat
Keyword(s):  
Coupled Oscillators ◽  
Dynamical Behavior ◽  
Negative Stiffness ◽  
Nonlinear Frequency ◽  
Nonlinear Stiffness ◽  
Nonlinear Coupling ◽  
Nonlinear Dynamical ◽  
Wide Range ◽  
Coupling Force ◽  
Linear And Nonlinear

Abstract The linear and nonlinear stiffness coupling forces in dynamical oscillators are usually dominated by positive stiffness components. Therefore, plotting the resultant force in y-axis with respect to the change in displacement in x-axis results in an odd symmetry in the first and third quadrants of the xy-plane. However, the appearance of negative stiffness content in coupling elements between dynamical oscillators generates a force that can be dominated by an odd symmetry in the second and fourth quadrants. The underlying nonlinear dynamical behavior of systems employing this kind of force has not been well-studied in the literature. Accordingly, the considered system here is composed of two linear oscillators that are nonlinearly coupled by a force of which the negative stiffness content is dominant. Therefore, the underlying dynamical behavior of the considered system in physical and dimensionless forms is studied on the frequency-energy plots where many backbone curves of periodic solution have been obtained. It is found that within a wide range of nonlinear frequency levels, the nonlinear coupling force is dominated by a strong negative stiffness content at the obtained frequency-energy plots backbones.

Modal Damping and Frequency Variations in Nonlinearly Coupled Oscillators With Negative Linear Stiffness Components

2018 ◽  
Author(s):  
Fatima K. Alhammadi ◽  
Mohammad A. AL-Shudeifat
Keyword(s):  
Coupled Oscillators ◽  
Nonlinear Dynamical Systems ◽  
Equations Of Motion ◽  
Original System ◽  
Nonlinear Frequency ◽  
Nonlinear Coupling ◽  
Frequency Content ◽  
Damping Coefficients ◽  
Modal Damping ◽  
Coupling Force

A method is applied here to extract the amplitude-dependent modal damping coefficients and frequencies of nonlinearly coupled oscillators with a nonlinear force in which a negative linear stiffness is incorporated. The proposed method can be directly applied into the equations of motion of the original system where the solution is not required to be obtained a priori. The exact nonlinear frequency content in the nonlinear coupling element is employed to obtain an equivalent amplitude-dependent stiffness element using a scaling parameter that preserves the exact frequency content in the original nonlinear element. Therefore, at each amplitude in the nonlinear coupling force, the modal damping coefficients and frequencies are calculated from the eigensolution of the instantaneous amplitude-dependent equivalent system. It is found that the modal damping content is strongly affected by the nonlinear frequency content where the modal damping coefficients become amplitude-dependent quantities. The obtained amplitude-dependent damping coefficients are plotted with respect to the potential energy of the nonlinear coupling force. The method is also applicable with larger degree-of-freedom nonlinear dynamical systems in which negative and non-negative linear stiffness components are incorporated in the nonlinear coupling forces. The amplitude-dependent modal damping matrices of the amplitude-dependent equivalent systems are found to be satisfying all matrix similarity conditions with the linear modal damping matrix of the original system.

Analysis and Experimental Estimation of Nonlinear Dispersion in a Periodic String

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
Kevin L. Manktelow ◽  
Michael J. Leamy ◽  
Massimo Ruzzene
Keyword(s):  
Dispersion Curve ◽  
Wave Dispersion ◽  
Lumped Parameter Model ◽  
Nonlinear Frequency ◽  
Nonlinear Dispersion ◽  
Frequency Shifts ◽  
Wave Transport ◽  
Lumped Parameter ◽  
Dispersion Frequency ◽  
Nonlinear Frequency Response

Wave dispersion in a string carrying periodically distributed masses is investigated analytically and experimentally. The effect of the string's geometric nonlinearity on its wave propagation characteristics is analyzed through a lumped parameter model yielding coupled Duffing oscillators. Dispersion frequency shifts are predicted that correspond to the hardening behavior of the nonlinear chain and that relate well to the backbone of individual Duffing oscillators. Experiments conducted on a string of finite length illustrate the relation between measured resonances and the dispersion properties of the medium. Specifically, the locus of resonance peaks in the frequency/wavenumber domain outlines the dispersion curve and highlights the existence of a frequency bandgap. Moreover, amplitude-dependent resonance shifts induced by the string nonlinearity confirm the hardening characteristics of the dispersion curve. Analytical and experimental results provide a critical link between nonlinear dispersion frequency shifts and the backbone curves intrinsic to nonlinear frequency response functions. Moreover, the study confirms that amplitude-dependent wave properties for nonlinear periodic systems may be exploited for tunability of wave transport characteristics such as frequency bandgaps and wave speeds.

Transition from a coherent three wave system to turbulence with application to the fluid closure

2014 ◽  
Vol 81 (1) ◽  
Author(s):  
Jan Weiland ◽  
Chuan S. Liu ◽  
Anatoly Zagorodny
Keyword(s):  
Distribution Function ◽  
Wave Packet ◽  
Random Phase ◽  
Maxwellian Distribution ◽  
Wave System ◽  
Nonlinear Frequency ◽  
Frequency Shifts ◽  
Wave Dynamics ◽  
Phase Velocities ◽  
Maxwellian Distribution Function

AbstractWe start from a Mattor–Parker system and its generalization to include diffusion and derive the Random Phase equations. It is shown that the same type of fluid closure holds in the coherent and turbulent regimes. This is due to the fact that the Random Phase levels (1/I1 = 1/I2 + 1/I3), where Ij is the intensity of wave packet ‘j’, are attractors for the wave dynamics both in the coherent and incoherent cases. Focus here is on the wave dynamics with phase velocities varying due to nonlinear frequency shifts. Thus a Maxwellian distribution function is kept in all cases.

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