Analytical Treatment for Bistable Nonlinearly Coupled Oscillators

2017 ◽  
Author(s):  
Mohammad A. AL-Shudeifat ◽  
Adnan S. Saeed
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Coupled Oscillators ◽  
Normal Modes ◽  
Nonlinear Coupling ◽  
Analytical Treatment ◽  
Internal Resonances ◽  
Restoring Force ◽  
Forcing Function ◽  
Coupling Force

Here, we introduce an analytical approximation to the exact solution of a bistable nonlinearly coupled oscillators (NLC-LOs) to study the internal resonance at the nonlinear normal modes (NNMs). The considered system is composed of two symmetrical linear oscillators coupled by a bistable nonlinear coupling restoring force. The coupling restoring force includes negative and nonnegative linear and nonlinear stiffness components. The introduced approximate analytical solution for the considered bistable NLC-LOs system is mainly proposed for the cases of which the exact frequency and the exact solution are neither available nor valid. The proposed solution depends on the application of the local equivalent linear stiffness method (LELSM) to linearize the nonlinear coupling force according to the non-linear frequency content in the original system. Accordingly, the bistable nonlinear coupling force in the NLC-LOs is replaced by an equivalent periodic forcing function of which the frequency is equal to that of the original NLC-LOs system. Therefore, the original NLC-LOs system is decoupled into two forced single degree-of-freedom subsystems where the analytical solution can be directly obtained. This obtained analytical solution is found to be highly accurate approximation for the exact solution, especially at internal resonances that occur on some NNMs of the system.

Analytical Solution of Two Coupled Oscillators With a Nonlinear Coupling Resorting Force

2014 ◽  
Author(s):  
Mohammad A. Al-Shudeifat ◽  
Thomas D. Burton
Keyword(s):  
Analytical Solution ◽  
Nonlinear Dynamical System ◽  
Degree Of Freedom ◽  
Nonlinear Frequency ◽  
Nonlinear Coupling ◽  
Restoring Force ◽  
Nonlinear Dynamical ◽  
Forcing Function ◽  
Local Equivalent ◽  
Coupling Force

An approach for accurate analytical solution of a two degree-of-freedom nonlinear dynamical system coupled with a strongly nonlinear restoring force is presented here. The approach is based on the application of the local equivalent linear stiffness method (LELSM) to linearize the nonlinear coupling stiffness in the system based on the nonlinear frequency calculation. Consequently, the system can be decoupled into two forced single degree-of-freedom subsystems by replacing the nonlinear coupling force with a forcing function where the solution can be analytically obtained. Different combinations of the positive and negative linear and cubic stiffness components are considered in the nonlinear coupling force. For all considered stiffness combinations, the obtained analytical solution strongly agrees with the numerical simulation of the system. In addition, the internal resonance is found not to significantly affect the accuracy of the analytical solution.

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.

Modal Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes*

2002 ◽  
Vol 124 (2) ◽  
pp. 229-236 ◽  
Author(s):  
Eric Pesheck ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Degrees Of Freedom ◽  
Invariant Manifolds ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Nonlinear Coupling ◽  
Rotating Beam ◽  
Internal Resonances ◽  
Modal Expansion ◽  
Nonlinear Normal

A method for determining reduced-order models for rotating beams is presented. The approach is based on the construction of nonlinear normal modes that are defined in terms of invariant manifolds that exist for the system equations of motion. The beam considered is an idealized model for a rotor blade whose motions are dominated by transverse vibrations in the direction perpendicular to the plane of rotation (known as flapping). The mathematical model for the rotating beam is relatively simple, but contains the nonlinear coupling that exists between transverse and axial deflections. When one employs standard modal expansion or finite element techniques to this system, this nonlinearity causes slow convergence, leading to models that require many degrees of freedom in order to achieve accurate dynamical representations. In contrast, the invariant manifold approach systematically accounts for the nonlinear coupling between linear modes, thereby providing models with very few degrees of freedom that accurately capture the essential dynamics of the system. Models with one and two nonlinear modes are considered, the latter being able to handle systems with internal resonances. Simulation results are used to demonstrate the validity of the approach and to exhibit features of the nonlinear modal responses.

Nonlinear Energy Sink With a Non-Traditional Kind of Nonlinear Restoring Force

2015 ◽  
Author(s):  
Mohammad A. Al-Shudeifat
Keyword(s):  
Nonlinear Energy Sink ◽  
Initial Energy ◽  
Type I ◽  
Nonlinear Coupling ◽  
Restoring Force ◽  
Nonlinear Restoring Force ◽  
Coupling Force ◽  
Restoring Forces ◽  
Energy Sink ◽  
Nonlinear Energy

Enhanced nonlinear energy sink (NES) is addressed here by employing a non-traditional kind of a nonlinear restoring force. The usual nonlinear coupling element between the NES and the linear oscillator (LO) in the literature generates essentially nonlinear restoring force between the NES and the LO. Unlike Type I NES, here the nonlinear coupling force has varying components during the oscillation which appear in closed loops under the effect of damping terms. This NES attachment with the LO rapidly absorbs and immediately dissipates significant portion of the initial energy induced into the LO through a strong resonance capture between the NES and LO responses. The proposed design could also be promising for energy harvesting purposes. The obtained results by numerical simulation show that employing this type of nonlinear restoring force for passive targeted energy transfer (TET) is more promising than some other types of NESs in which purely cubic stiffness restoring forces have been incorporated.

scholarly journals Asymptotic Numerical Method and Polynomial Chaos Expansion for the Study of Stochastic Non-Linear Normal Modes

2015 ◽  
Author(s):  
Alfonso M. Panunzio ◽  
Loïc Salles ◽  
Christoph Schwingshackl ◽  
Muzio Gola
Keyword(s):  
Resonance Frequency ◽  
Numerical Method ◽  
Normal Modes ◽  
Nonlinear Dynamic Analysis ◽  
Computationally Efficient ◽  
Internal Resonances ◽  
Mass System ◽  
Asymptotic Numerical Method ◽  
Forcing Function ◽  
Non Linear

Nonlinear normal mode (NNM) analysis is one emerging technique to analyse the nonlinear vibration of bladed-disk. It links the resonance frequency to the energy present in the system, and allows a simple identification of internal resonances in the structure. Non-linear vibration analysis is traditionally carried out under the assumption that the mechanical properties and forcing function are deterministic. Since every mechanical system is by nature uncertain a truly accurate nonlinear dynamic analysis requires the inclusion of random variables in the response predictions. The propagation of random input uncertainties in a NNM analysis is the main aim of the presented work. The Asymptotic Numerical Method (ANM) will be used to calculate the NNMs for a contact problem in a computationally efficient way. The stochastic NNM permits to quantify the effect of uncertainties on the resonance frequency and the change in mode shape due to non-linearities, leading to the calculation of uncertain internal resonances. The proposed method is initially applied to a simple spring-mass system to demonstrate the effects of uncertainty on the NNM predictions. In a second step a blade-casing interaction with localized contact non-linearity is investigated with a real geometry. The resulting NNMs show the presence of internal resonance for both cases.

FREQUENCY SYNCHRONIZATION IN NETWORKS OF COUPLED OSCILLATORS, A MONOTONE DYNAMICAL SYSTEMS APPROACH

2009 ◽  
Vol 19 (12) ◽  
pp. 4107-4116 ◽  
Author(s):  
WEN-XIN QIN
Keyword(s):  
Dynamical Systems ◽  
Coupling Strength ◽  
Coupled Oscillators ◽  
Systems Approach ◽  
System Size ◽  
Coupling Scheme ◽  
Frequency Synchronization ◽  
Nonlinear Coupling ◽  
New Approach ◽  
Monotone Dynamical Systems

We propose a new approach to investigate the frequency synchronization in networks of coupled oscillators. By making use of the theory of monotone dynamical systems, we show that frequency synchronization occurs in networks of coupled oscillators, provided the coupling scheme is symmetric, connected, and strongly cooperative. Our criterion is independent of the system size, the coupling strength and the details of the connections, and applies also to nonlinear coupling schemes.

Nonlinear Dynamics of a System of Coupled Oscillators With Essential Stiffness Nonlinearities

2003 ◽  
Author(s):  
Alexander F. Vakakis ◽  
Richard H. Rand
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Resonance Capture ◽  
Periodic Motions ◽  
Internal Resonances ◽  
Dynamical Mechanism ◽  
Elliptic Orbits ◽  
Weakly Coupled ◽  
Resonant Dynamics ◽  
Nonlinear Normal

We study the resonant dynamics of a two-degree-of-freedom system composed a linear oscillator weakly coupled to a strongly nonlinear one, with an essential (nonlinearizable) cubic stiffness nonlinearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (nonlinear normal modes — NNMs), as well as, asynchronous periodic motions (elliptic orbits — EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets ‘captured’ in the neighborhood of a damped NNM before ‘escaping’ and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive nonlinear energy pumping phenomena from the linear to the nonlinear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.

Nonlinear Normal Modes of Buckled Beams: Three-to-One and One-to-One Internal Resonances

1997 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Walter Lacarbonara ◽  
Char-Ming Chin
Keyword(s):  
Boundary Conditions ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Internal Resonances ◽  
Buckling Mode ◽  
Stable Mode ◽  
One To One ◽  
Nonlinear Normal

Abstract Nonlinear normal modes of a buckled beam about its first buckling mode shape are investigated. Fixed-fixed boundary conditions are considered. The cases of three-to-one and one-to-one internal resonances are analyzed. Approximate expressions for the nonlinear normal modes are obtained by applying the method of multiple scales to the governing integro-partial-differential equation and boundary conditions. Curves displaying variation of the amplitude with the internal resonance detuning parameter are generated. It is shown that, for a three-to-one internal resonance between the first and third modes, the beam may possess either one stable mode, or three stable normal modes, or two stable and one unstable normal modes. On the other hand, for a one-to-one internal resonance between the first and second modes, two nonlinear normal modes exist. The two nonlinear modes are either neutrally stable or unstable. In the case of one-to-one resonance between the third and fourth modes, two neutrally stable, nonlinear normal modes exist.

scholarly journals Spectroscopy of z=0 Lifshitz Black Hole

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Gulnihal Tokgoz ◽  
Izzet Sakalli
Keyword(s):  
Exact Solution ◽  
Black Hole ◽  
Normal Modes ◽  
Hawking Temperature ◽  
Massive Scalar ◽  
Local Mass ◽  
Entropy Formula ◽  
Adiabatic Invariant Quantity ◽  
Lifshitz Black Hole ◽  
Area Spectra

We studied the thermodynamics and spectroscopy of a 4-dimensional, z=0 Lifshitz black hole (Z0LBH). Using Wald’s entropy formula and the Hawking temperature, we derived the quasi-local mass of the Z0LBH. Based on the exact solution to the near-horizon Schrödinger-like equation (SLE) of the massive scalar waves, we computed the quasi-normal modes of the Z0LBH via employing the adiabatic invariant quantity for the Z0LBH. This study shows that the entropy and area spectra of the Z0LBH are equally spaced.

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