Intrinsic Localized Modes of Harmonic Oscillations in Nonlinear Oscillator Arrays

2013 ◽  
Vol 8 (4) ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Keisuke Nishimura
Keyword(s):  
Frequency Response ◽  
Initial Conditions ◽  
Excitation Frequency ◽  
Basins Of Attraction ◽  
Localized Modes ◽  
Response Curves ◽  
Harmonic Oscillations ◽  
Intrinsic Localized Modes ◽  
Mdof Systems ◽  
Oscillator Arrays

Intrinsic localized modes (ILMs) are investigated in an array with N Duffing oscillators that are weakly coupled with each other when each oscillator is subjected to sinusoidal excitation. The purpose of this study is to investigate the behavior of ILMs in nonlinear multi-degree-of-freedom (MDOF) systems. In the theoretical analysis, van der Pol's method is employed to determine the expressions for the frequency response curves for fundamental harmonic oscillations. In the numerical calculations, the frequency response curves are shown for N = 2 and 3 and compared with the results of the numerical simulations. Basins of attraction are shown for a two-oscillator array with hard-type nonlinearities to examine the possibility of appearance of ILMs when an oscillator is disturbed. The influences of the connecting springs for both hard- and soft-type nonlinearities on the appearance of the ILMs are examined. Increasing the values of the connecting spring constants may cause Hopf bifurcation followed by amplitude modulated motion (AMM) including chaotic vibrations. The influence of the imperfection of an oscillator is also investigated. Bifurcation sets are calculated to show the influence of the system parameters on the excitation frequency range of ILMs. Furthermore, time histories are shown for the case of N = 10, and many patterns of ILMs may appear depending on the initial conditions.

Intrinsic Localized Modes of Harmonic Oscillations in Pendulum-Arrays Subjected to Horizontal Excitation

2013 ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Keisuke Nishimura
Keyword(s):  
Theoretical Analysis ◽  
Frequency Response ◽  
Excitation Frequency ◽  
System Response ◽  
Stable Steady State ◽  
Frequency Range ◽  
Localized Modes ◽  
Response Curves ◽  
Harmonic Oscillations ◽  
Intrinsic Localized Modes

The behavior of intrinsic localized modes (ILMs) is investigated for an array with N pendula which are connected with each other by weak, linear springs when the array is subjected to horizontal, sinusoidal excitation. In the theoretical analysis, van der Pol’s method is employed to determine the expressions for the frequency response curves for fundamental harmonic oscillations. In the numerical calculations, the frequency response curves are presented for N = 2 and 3 and compared with the results of the numerical simulations. Patterns of oscillations are classified according to the stable steady-state solutions of the response curves, and the patterns in which ILMs appear are discussed in detail. The influence of the connecting springs of the pendula on the appearance of ILMs is examined. Increasing the values of the connecting spring constants may affect the excitation frequency range of ILMs and cause Hopf bifurcation to occur, followed by amplitude modulated motions (AMMs) including chaotic vibrations. The influence of the imperfections of the pendula on the system response is also investigated. Bifurcation sets are calculated to examine the influence of the system parameters on the excitation frequency range of ILMs and determine the threshold value for the connecting spring constant after which ILMs do not appear. Experiments were conducted for N = 2, and the data were compared with the theoretical results in order to confirm the validity of the theoretical analysis.

Intrinsic Localized Modes of Harmonic Oscillations in Pendulum Arrays Subjected to Horizontal Excitation

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Keisuke Nishimura
Keyword(s):  
Theoretical Analysis ◽  
Frequency Response ◽  
Excitation Frequency ◽  
System Response ◽  
Stable Steady State ◽  
Frequency Range ◽  
Localized Modes ◽  
Response Curves ◽  
Harmonic Oscillations ◽  
Intrinsic Localized Modes

The behavior of intrinsic localized modes (ILMs) is investigated for an array with N pendula which are connected with each other by weak, linear springs when the array is subjected to horizontal, sinusoidal excitation. In the theoretical analysis, van der Pol's method is employed to determine the expressions for the frequency response curves for fundamental harmonic oscillations. In the numerical calculations, the frequency response curves are presented for N = 2 and 3 and compared with the results of the numerical simulations. Patterns of oscillations are classified according to the stable steady-state solutions of the response curves, and the patterns in which ILMs appear are discussed in detail. The influence of the connecting springs of the pendula on the appearance of ILMs is examined. Increasing the values of the connecting spring constants may affect the excitation frequency range of ILMs and cause Hopf bifurcation to occur, followed by amplitude modulated motions (AMMs) including chaotic vibrations. The influence of the imperfections of the pendula on the system response is also investigated. Bifurcation sets are calculated to examine the influence of the system parameters on the excitation frequency range of ILMs and determine the threshold value for the connecting spring constant above which ILMs do not appear. Experiments were conducted for N = 2, and the data were compared with the theoretical results in order to confirm the validity of the theoretical analysis.

Intrinsic Localized Modes of Principal Parametric Resonances in Pendulum Arrays Subjected to Vertical Excitation

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Chongyue Shi ◽  
Keisuke Nishimura
Keyword(s):  
Experimental Data ◽  
Theoretical Analysis ◽  
Frequency Response ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Localized Modes ◽  
Response Curves ◽  
Chaotic Motions ◽  
Intrinsic Localized Modes ◽  
Spring Constants

Intrinsic localized modes (ILMs) are investigated in an N-pendulum array subjected to vertical harmonic excitation. The pendula behave nonlinearly and are coupled with each other because they are connected by torsional, weak, linear springs. In the theoretical analysis, van der Pol's method is employed to determine the expressions for frequency response curves for the principal parametric resonance, considering the nonlinear restoring moment of the pendula. In the numerical results, frequency response curves for N = 2 and 3 are shown to examine the patterns of ILMs, and demonstrate the influences of the connecting spring constants and the imperfections of the pendula. Bifurcation sets are also calculated to show the excitation frequency range and the conditions for the occurrence of ILMs. Increasing the connecting spring constants results in the appearance of Hopf bifurcations. The numerical simulations reveal the occurrence of ILMs with amplitude modulated motions (AMMs), including chaotic motions. ILMs were observed in experiments, and the experimental data were compared with the theoretical results. The validity of the theoretical analysis was confirmed by the experimental data.

Modal Analysis to Interpret Localization Phenomena of Harmonic Oscillations in Nonlinear Oscillator Arrays

2019 ◽  
Author(s):  
Yuji Harata ◽  
Takashi Ikeda
Keyword(s):  
Coordinate System ◽  
Frequency Response ◽  
Equations Of Motion ◽  
Harmonic Excitation ◽  
Vibrational Modes ◽  
Response Curves ◽  
Internal Resonances ◽  
Harmonic Oscillations ◽  
Oscillator Arrays ◽  
Phase Angles

Abstract This paper investigates localization phenomena in a nonlinear array with N Duffing oscillators connected by weak, linear springs when the array is subjected to harmonic excitation. In the theoretical analysis, the equations of motion are derived for: (1) the physical coordinate system, and (2) modal coordinate system. The modal equations of motion form an autoparametric system, i.e., the excitation acts directly on the first mode of vibration, and the other modes are indirectly excited because they are nonlinearly coupled with the first mode. Van der Pol’s method is employed to obtain the solutions of the harmonic oscillations, and then the expressions of the frequency response curves are given. In the numerical calculations, the frequency response curves of the amplitudes and phase angles in the cases of N = 2 and 3 are presented. The frequency response curves, obtained in the modal coordinate system, demonstrate that localization phenomena occur in the physical coordinate system when multiple vibrational modes simultaneously appear. When imperfections exist in the N Duffing oscillators, the modal equations of motion do not form an autoparametric system because the external excitation directly acts on all modes. Instead, internal resonances may occur in such systems.

Sensitivity Analysis of the Frequency Response of a Piecewise Linear System in a Frequency Island

2004 ◽  
Vol 10 (2) ◽  
pp. 175-198 ◽  
Author(s):  
A Narimani ◽  
M Farid Golnaraghi ◽  
G Nakhaie Jazar
Keyword(s):  
Frequency Response ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Excitation Frequency ◽  
Relative Displacement ◽  
System Parameters ◽  
Dynamical Characteristics ◽  
Piecewise Linear System ◽  
Amplitude Vibration ◽  
Large Amplitude Vibration

In this paper, we analyze the frequency response of a piecewise linear suspension system. Dynamical characteristics of the suspension will suddenly change when its relative displacement exceeds a clearance. Piecewise linear characteristics occur, for example, wherever we use stoppers to prevent the system from excessive relative displacement. A modified averaging method is used to find the frequency response of the system to a harmonic base excitation. A frequency island is observed, which corresponds to large amplitude vibration for a certain range of system parameters. The island is an isolated region that cannot be reached by the variation of excitation frequency and depends upon initial conditions. On the frequency island, the isolator amplifies the amplitude of vibration rather than suppressing it. This will be dangerous in applications where stoppers are installed to ensure relative displacement is limited. The ranges of system parameters causing the frequency island are determined. The results obtained by an analytical method are verified using numerical simulation.

Nonlinear Response of Short Squeeze Film Dampers

1980 ◽  
Vol 102 (1) ◽  
pp. 51-58 ◽  
Author(s):  
D. L. Taylor ◽  
B. R. K. Kumar
Keyword(s):  
Frequency Response ◽  
Phase Angle ◽  
Transient Response ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Polar Coordinates ◽  
Squeeze Film ◽  
Response Curves ◽  
Squeeze Film Damper ◽  
Fluid Forces

This paper considers the methodology of numerical integration for prediction of dynamic response of squeeze film damper systems. A planar rotor carried in a squeeze film damper with linear centering spring is considered. Governing differential equations are expressed in polar coordinates, and fluid forces are obtained from the Ocvirk short bearing integrals. The rotating unbalance response is presented as a function of speed, unbalance, and a bearing parameter. Runge Kutta integration techniques are used to obtain numerical solutions for transient response and frequency response. The 2π film approximation results in almost linear frequency response curves. However, the π film response is very nonlinear, demonstrating the well known multiple valued response and associated hardening jump/drop phenomenon. The π film transient response is analyzed within the speed range of bistable operation to determine the effects of initial conditions, the domains of convergence, and the relative strengths of stability of each solution. The transient response is found to be most sensitive to initial values of phase angle and phase angle velocity. Initial eccentricity and eccentric velocity are much less important. In general, of the two steady state solutions, the one with lower eccentricity appears to be more stable, with a larger domain of convergence. Examples show how premature termination of the integration can lead to erroneous conclusions.

Efficacy of a Sliding Beam as a Nonlinear Vibration Isolator

2007 ◽  
Author(s):  
R. A. Ibrahim ◽  
R. J. Somnay
Keyword(s):  
Friction Coefficient ◽  
Initial Conditions ◽  
Excitation Frequency ◽  
Random Excitation ◽  
Excitation Power ◽  
Basins Of Attraction ◽  
System Response ◽  
Density Level ◽  
Vibration Isolator ◽  
Power Spectral

The influence of friction due to beam sliding at its supports on its dynamic behavior and its efficacy as a nonlinear isolator is studied numerically under sinusoidal and random excitation excitations. Under sinusoidal excitation, the equation of motion of the system is solved numerically and the solution is utilized to estimate the system transmissibility. It is found that when the excitation frequency is increased beyond resonance, the friction at the sliding supports serves to improve the transmissibility. The dependence of the response on initial conditions establishes the basins of attraction for different values of friction coefficient and excitation frequency and amplitude. Under random excitation, the system response statistics are estimated from Monte Carlo simulation results for different values of friction coefficient and excitation power spectral density level. The friction is found to result in a significant reduction of the system response mean square.

Nonlinear Dynamic Responses of Elastic Structures With Two Rectangular Liquid Tanks Subjected to Horizontal Excitation

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Takashi Ikeda
Keyword(s):  
Frequency Response ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Pitchfork Bifurcation ◽  
Harmonic Excitation ◽  
Dynamic Responses ◽  
Response Curves ◽  
Chaotic Vibrations ◽  
Tuned Liquid Dampers ◽  
Theoretical Results

Nonlinear vibrations of an elastic structure with two partially filled liquid tanks subjected to horizontal harmonic excitation are investigated. The natural frequencies of the structure and sloshing satisfy the tuning condition 1:1:1 when tuned liquid dampers are used. The equations of motion for the structure and the modal equations of motion for the first, second, and third sloshing modes are derived by using Galerkin’s method, taking into account the nonlinearity of the sloshing. Then, van der Pol’s method is employed to determine the frequency response curves. It is found in calculating the frequency response curves that pitchfork bifurcation can occur followed by “localization phenomenon” for a specific excitation frequency range. During this range, sloshing occurs at different amplitudes in the two tanks, even if the dimensions of both tanks are identical. Furthermore, Hopf bifurcation may occur followed by amplitude- and phase-modulated motions including chaotic vibrations. In addition, Lyapunov exponents are calculated to prove the occurrence of both amplitude-modulated motions and chaotic vibrations. Bifurcation sets are also calculated to show the influence of the system parameters on the frequency response. Experiments were conducted to confirm the validity of the theoretical results. It was found that the theoretical results were in good agreement with the experimental data.

BIFURCATION STRUCTURE AT 1/3-SUBHARMONIC RESONANCE IN AN ASYMMETRIC NONLINEAR ELASTIC OSCILLATOR

1996 ◽  
Vol 06 (08) ◽  
pp. 1529-1546 ◽  
Author(s):  
G. REGA ◽  
A. SALVATORI
Keyword(s):  
Invariant Manifolds ◽  
Initial Conditions ◽  
Basins Of Attraction ◽  
Subharmonic Resonance ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Bifurcation Structure ◽  
Response Curves ◽  
Nonlinear Elastic ◽  
Insight Into

The attractor-basin bifurcation structure in an asymmetric nonlinear oscillator representative of the planar finite forced dynamics of elastic structural systems with initial curvature is studied at the 1/3-subharmonic resonance regime. Local and global analyses are made by means of different computational tools to obtain frequency-response curves of coexisting regular solutions, bifurcation diagrams ensuing from different sets of initial conditions, manifolds structure of direct and inverse saddles corresponding to unstable periodic solutions, basins of attraction at different values of the control parameter. Deep insight into the global dynamics of the system and its evolution is achieved through the analysis of synthetic attractor-basin-manifold phase portraits. The topological mechanisms which entail onset and disappearance of various attractors, and the main and secondary evolutions to chaos, are identified. Special attention is devoted to the analysis of sudden bifurcational events characterizing the system global dynamics, associated with the topological behavior of the invariant manifolds of several direct and inverse saddles. Features of basin metamorphosis, attractor-basin accessibility, and window occurrence are examined. The approach followed, consisting in combined bifurcation analysis of the attractor-basin structure and of the manifold structure, is thought to be useful for a variety of dynamical systems.

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