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.

scholarly journals Nonlinear Dynamics of a Duffing-Like Negative Stiffness Oscillator: Modeling and Experimental Characterization

2020 ◽  
Vol 2020 ◽  
pp. 1-13 ◽  
Author(s):  
D. Anastasio ◽  
A. Fasana ◽  
L. Garibaldi ◽  
S. Marchesiello
Keyword(s):  
Nonlinear Dynamics ◽  
Chaotic Motion ◽  
Nonlinear Oscillations ◽  
Dynamical Behavior ◽  
Shaking Table ◽  
Negative Stiffness ◽  
Nonlinear Dynamical ◽  
Current Collection ◽  
Input Amplitude ◽  
Linear And Nonlinear

In this paper, a negative stiffness oscillator is modelled and tested to exploit its nonlinear dynamical characteristics. The oscillator is part of a device designed to improve the current collection quality in railway overhead contact lines, and it acts like an asymmetric double-well Duffing system. Thus, it exhibits two stable equilibrium positions plus an unstable one, and the oscillations can either be bounded around one stable point (small oscillations) or include all the three positions (large oscillations). Depending on the input amplitude, the oscillator can exhibit linear and nonlinear dynamics and chaotic motion as well. Furthermore, its design is asymmetrical, and this plays a key role in its dynamic response, as the two natural frequencies associated with the two stable positions differ from each other. The first purpose of this study is to understand the dynamical behavior of the system in the case of linear and nonlinear oscillations around the two stable points and in the case of large oscillations associated with a chaotic motion. To accomplish this task, the device is mounted on a shaking table and it is driven with several levels of excitations and with both harmonic and random inputs. Finally, the nonlinear coefficients associated with the nonlinearities of the system are identified from the measured data.

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.

scholarly journals Experimental Characterization of Friction in a Negative Stiffness Nonlinear Oscillator

Vibration ◽  
2020 ◽  
Vol 3 (2) ◽  
pp. 132-148
Author(s):  
Dario Anastasio ◽  
Stefano Marchesiello
Keyword(s):  
Dynamical Behavior ◽  
Harmonic Balance Method ◽  
Nonlinear Behavior ◽  
Nonlinear Damping ◽  
Negative Stiffness ◽  
Experimental Characterization ◽  
Restoring Force ◽  
Wide Range ◽  
Energy Dissipated

Nonlinear dissipative phenomena are common features of many dynamical systems and engineering applications, and their experimental characterization has always been a challenge among the research community. Within the wide range of nonlinear damping mechanisms, friction is surely one of the most common, and with a high impact on the dynamical behavior of structures. In this paper, the nonlinear identification of friction in a negative stiffness oscillator is pursued. The structure exhibits a strong nonlinear behavior, mainly due to its polynomial elastic restoring force with a negative stiffness region. This leads to an asymmetric double-well potential with two stable equilibrium positions, and the possibility of switching between them in a chaotic way. Friction plays a crucial role in this context, as it derives from the continuous sliding between the central guide and the moving mass. The system is driven through harmonic tests with several input amplitudes, in order to estimate the variations in the energy dissipated per cycle. The identification of the frictional behavior is then pursed by minimizing the errors between the experimental measurements and the model predictions, using the harmonic balance method in conjunction with a continuation technique on the forcing amplitudes.

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.

scholarly journals Dynamic Analysis of Stockbridge Damper

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Nilson Barbieri ◽  
Renato Barbieri
Keyword(s):  
Loss Factor ◽  
Dynamical Behavior ◽  
Excitation Frequency ◽  
Vibration Signals ◽  
Nonlinear Dynamical ◽  
Nonlinear Mathematical Models ◽  
Linear And Nonlinear ◽  
Base Displacement ◽  
Two Parameters ◽  
Stockbridge Damper

In this work a machine cam with five different profiles was used to investigate the linear and nonlinear dynamical behavior of asymmetric Stockbridge damper with excitation frequencies in the range of 5≤f≤17 Hz. The experimental vibration signals were acquired through accelerometers placed along the sample. The loss factor and the Young's modulus were estimated through approximation of the experimental and numerical results using Genetic Algorithms (GAs). Linear and nonlinear mathematical models were used to adjust the data. The two parameters are dependent on the excitation frequency and the amplitude of the base displacement. The results are validated comparing typical impedance curves obtained in conventional testing using an electromechanical shaker.

Asymptotic for LS estimators in the EV regression model for dependent errors

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4845-4856
Author(s):  
Konrad Furmańczyk
Keyword(s):  
Regression Model ◽  
Functional Dependence ◽  
Dependent Data ◽  
Errors In Variables ◽  
Mixing Conditions ◽  
Asymptotic Results ◽  
Dependence Measure ◽  
Wide Range ◽  
Ev Regression Model ◽  
Linear And Nonlinear

We study consistency and asymptotic normality of LS estimators in the EV (errors in variables) regression model under weak dependent errors that involve a wide range of linear and nonlinear time series. In our investigations we use a functional dependence measure of Wu [16]. Our results without mixing conditions complete the known asymptotic results for independent and dependent data obtained by Miao et al. [7]-[10].

scholarly journals Bistable dynamics beyond rotating wave approximation

2014 ◽  
Vol 23 (02) ◽  
pp. 1450019 ◽  
Author(s):  
Y. A. Sharaby ◽  
S. Lynch ◽  
A. Joshi ◽  
S. S. Hassan
Keyword(s):  
Ring Cavity ◽  
Dynamical Behavior ◽  
Single Mode ◽  
Unstable State ◽  
Nonlinear Dynamical ◽  
Rotating Wave Approximation ◽  
Wave Approximation ◽  
Atomic Medium ◽  
Bistable Dynamics ◽  
Rotating Wave

In this paper, we investigate the nonlinear dynamical behavior of dispersive optical bistability (OB) for a homogeneously broadened two-level atomic medium interacting with a single mode of the ring cavity without invoking the rotating wave approximation (RWA). The periodic oscillations (self-pulsing) and chaos of the unstable state of the OB curve is affected by the counter rotating terms through the appearance of spikes during its periods. Further, the bifurcation with atomic detuning, within and outside the RWA, shows that the OB system can be converted from a chaotic system to self-pulsing system and vice-versa.

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