Resonance behavior in coupled Van der Pol harmonic oscillators with controllers and delayed feedback

2020 ◽  
pp. 107754632093818
Author(s):  
Ashraf T EL-Sayed
Keyword(s):  
Frequency Response ◽  
Time Delays ◽  
Degrees Of Freedom ◽  
Primary Resonance ◽  
Delayed Feedback ◽  
Harmonic Oscillators ◽  
Feedback Signal ◽  
Response Curves ◽  
Internal Resonances ◽  
Van Der Pol

It has been revealed in the proposed work that a pair of delay positive position feedback control can lessen the vibration response of double Van der Pol oscillators with external forces. We also studied the effects of both the control and the delayed feedback signal gains to illustrate the low vibration amplitudes. The averaging perturbation process has been used to consider the frequency-response equations of amplitudes and modulation phases at the primary resonance and one-to-one internal resonances. According to the perturbation solutions for the four-degrees-of-freedom system, we presented the frequency response curves that were periodic in the time delays. The stability analysis presented in this study has shown optimum stable ranges. If the time delays increase, the steady-state amplitudes of the oscillator’s system will periodically result in few stable regions and more unstable ones. The numerical simulation has been introduced to check the analytical approximation. It was also found to be almost identical after presenting the comparison of the results.

Effects of nonlinear interactions of flexural modes on the performance of a beam autoparametric vibration absorber

2019 ◽  
Vol 26 (7-8) ◽  
pp. 459-474
Author(s):  
Saeed Mahmoudkhani ◽  
Hodjat Soleymani Meymand
Keyword(s):  
Frequency Response ◽  
Internal Resonance ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Vibration Absorber ◽  
Primary System ◽  
Lumped Mass ◽  
Response Curves ◽  
Internal Resonances ◽  
The One

The performance of the cantilever beam autoparametric vibration absorber with a lumped mass attached at an arbitrary point on the beam span is investigated. The absorber would have a distinct feature that in addition to the two-to-one internal resonance, the one-to-three and one-to-five internal resonances would also occur between flexural modes of the beam by tuning the mass and position of the lumped mass. Special attention is paid on studying the effect of these resonances on increasing the effectiveness and extending the range of excitation amplitudes at which the autoparametric vibration absorber remains effective. The problem is formulated based on the third-order nonlinear Euler–Bernoulli beam theory, where the assumed-mode method is used for deriving the discretized equations of motion. The numerical continuation method is then applied to obtain the frequency response curves and detect the bifurcation points. The harmonic balance method is also employed for detecting the type of internal resonances between flexural modes by inspecting the frequency response curves corresponding to different harmonics of the response. Parametric studies on the performance of the absorber are conducted by varying the position and mass of the lumped mass, while the frequency ratio of the primary system to the first mode of the beam is kept equal to two. Results indicated that the one-to-five internal resonance is especially responsible for the considerable enhancement of the performance.

Nonlinear Finite Element-Based Path Following of Periodic Solutions

2011 ◽  
Author(s):  
Andrea Arena ◽  
Giovanni Formica ◽  
Walter Lacarbonara ◽  
Harry Dankowicz
Keyword(s):  
Finite Element ◽  
Vector Field ◽  
Frequency Response ◽  
Periodic Solutions ◽  
Parametric Resonance ◽  
Primary Resonance ◽  
Path Following ◽  
State Variables ◽  
Response Curves ◽  
Space Dependence

A computational framework is proposed to path follow the periodic solutions of nonlinear spatially continuous systems and more general coupled multiphysics problems represented by systems of partial differential equations with time-dependent excitations. The set of PDEs is cast in first order differential form (in time) u˙ = f(u,s,t;c) where u(s,t) is the vector collecting all state variables including the velocities/time rates, s is a space coordinate (here, one-dimensional systems are considered without lack of generality for the space dependence) and t denotes time. The vector field f depends, in general, not only on the classical state variables (such as positions and velocities) but also on the space gradients of the leading unknowns. The space gradients are introduced as part of the state variables. This is justified by the mathematical and computational requirements on the continuity in space up to the proper differential order of the space gradients associated with the unknown position vector field. The path following procedure employs, for the computation of the periodic solutions, only the evaluation of the vector field f. This part of the path following procedure within the proposed combined scheme was formerly implemented by Dankowicz and coworkers in a MATLAB software package called COCO. The here proposed procedure seeks to discretize the space dependence of the variables using finite elements based on Lagrangian polynomials which leads to a discrete form of the vector field f. A concurrent bifurcation analysis is carried out by calculating the eigenvalues of the monodromy matrix. A hinged-hinged nonlinear beam subject to a primary-resonance harmonic transverse load or to a parametric-resonance horizontal end displacement is considered as a case study. Some primary-resonance frequency-response curves are calculated along with their stability to assess the convergence of the discretization scheme. The frequency-response curves are shown to be in close agreement with those calculated by direct integration of the PDEs through the FE software called COMSOL Multiphysics. Besides primary-resonance direct forcing conditions, also parametric forcing causing the principal parametric resonance of the lowest two bending modes is considered through construction of the associated transition curves. The proposed approach integrates algorithms from the finite element and bifurcation domains thus enabling an accurate and effective unfolding of the bifurcation and post-bifurcation scenarios of nonautonomous PDEs with the underlying structures.

Nonlinear Two-to-One Internal Resonance for Broadband Energy Harvesting

2015 ◽  
Author(s):  
D. X. Cao ◽  
S. Leadenham ◽  
A. Erturk
Keyword(s):  
Energy Harvesting ◽  
Frequency Response ◽  
Internal Resonance ◽  
Numerical Solutions ◽  
Primary Resonance ◽  
Frame Structure ◽  
Response Curves ◽  
Current Effort ◽  
Bandwidth Enhancement ◽  
Piezoelectric Coupling

The transformation of waste vibration energy into low-power electricity has been heavily researched to enable self-sustained wireless electronic components. Monostable and bistable nonlinear oscillators have been explored by several researchers in an effort to enhance the frequency bandwidth of operation. Linear two degree of freedom (2-DOF) configurations as well as combination of a nonlinear single-DOF harvester with a linear oscillator to constitute a nonlinear 2-DOF harvester have also been explored to develop broadband energy harvesters. In the present work, the concept of nonlinear internal resonance in a continuous frame structure is explored for broadband energy harvesting. The L-shaped beam-mass structure with quadratic nonlinearity was formerly studied in the nonlinear dynamics literature to demonstrate modal energy exchange and the saturation phenomenon when carefully tuned for two-to-one internal resonance. In the current effort, piezoelectric coupling is introduced, and electromechanical equations of the L-shaped energy harvester are employed to explore the primary resonance behaviors around the first and the second linear natural frequencies for bandwidth enhancement. Simulations using approximate analytical frequency response equations as well as time-domain numerical solutions reveal that 2-DOF configuration with quadratic and two-to-one internal resonance could extend the bandwidth enhancement capability. Both electrical power and shunted vibration frequency response curves of steady-state solutions are explored in detail. Effects of various electromechanical system parameters, such as piezoelectric coupling and load resistance, on the overall dynamics of the internal resonance energy harvesting system are reported.

scholarly journals Hybrid rayleigh–van der pol–duffing oscillator: Stability analysis and controller

2021 ◽  
pp. 146134842110264
Author(s):  
Chun-Hui He ◽  
Dan Tian ◽  
Galal M Moatimid ◽  
Hala F Salman ◽  
Marwa H Zekry
Keyword(s):  
Fixed Points ◽  
Multiple Scales ◽  
Duffing Oscillator ◽  
Primary Resonance ◽  
Time History ◽  
Phase Portraits ◽  
Response Curves ◽  
Van Der Pol ◽  
Stability Performance ◽  
Linearized Stability

The current study examines the hybrid Rayleigh–Van der Pol–Duffing oscillator (HRVD) with a cubic–quintic nonlinear term and an external excited force. The Poincaré–Lindstedt technique is adapted to attain an approximate bounded solution. A comparison between the approximate solution with the fourth-order Runge–Kutta method (RK4) shows a good matching. In case of the autonomous system, the linearized stability approach is employed to realize the stability performance near fixed points. The phase portraits are plotted to visualize the behavior of HRVD around their fixed points. The multiple scales method, along with a nonlinear integrated positive position feedback (NIPPF) controller, is employed to minimize the vibrations of the excited force. Optimal conditions of the operation system and frequency response curves (FRCs) are discussed at different values of the controller and the system parameters. The system is scrutinized numerically and graphically before and after providing the controller at the primary resonance case. The MATLAB program is employed to simulate the effectiveness of different parameters and the controller on the system. The calculations showed that NIPPF is the best controller. The validations of time history and FRC of the analysis as well as the numerical results are satisfied by making a comparison among them.

scholarly journals Coupled Nonlinear Dynamics of Geometrically Imperfect Shear Deformable Extensible Microbeams

2015 ◽  
Vol 11 (4) ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Hamed Farokhi
Keyword(s):  
Dynamic Response ◽  
Degrees Of Freedom ◽  
Fourier Transforms ◽  
Dynamical Behavior ◽  
Primary Resonance ◽  
Continuous System ◽  
High Dimensional ◽  
Response Curves ◽  
Shear Deformable ◽  
Discretized Model

This paper aims at analyzing the coupled nonlinear dynamical behavior of geometrically imperfect shear deformable extensible microbeams based on the third-order shear deformation and modified couple stress theories. Using Hamilton's principle and taking into account extensibility, the three nonlinear coupled continuous expressions are obtained for an initially slightly curved (i.e., a geometrically imperfect) microbeam, describing the longitudinal, transverse, and rotational motions. A high-dimensional Galerkin scheme is employed, together with an assumed-mode technique, in order to truncate the continuous system with an infinite number of degrees of freedom into a discretized model with sufficient degrees of freedom. This high-dimensional discretized model is solved by means of the pseudo-arclength continuation technique for the system at the primary resonance, and also by direct time-integration to characterize the dynamic response at a fixed forcing amplitude and frequency; stability analysis is conducted via the Floquet theory. Apart from analyzing the nonlinear resonant response, the linear natural frequencies are obtained via an eigenvalue analysis. Results are shown through frequency–response curves, force–response curves, time traces, phase-plane portraits, and fast Fourier transforms (FFTs). The effect of taking into account the length-scale parameter on the coupled nonlinear dynamic response of the system is also highlighted.

scholarly journals Nonlinear Dynamic Response of Ropeway Roller Batteries via an Asymptotic Approach

2021 ◽  
Vol 11 (20) ◽  
pp. 9486
Author(s):  
Andrea Arena
Keyword(s):  
Nonlinear Dynamic ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Resonance Excitation ◽  
Dynamic Features ◽  
Response Curves ◽  
Internal Resonances ◽  
The Third ◽  
Fold Bifurcation

The nonlinear dynamic features of compression roller batteries were investigated together with their nonlinear response to primary resonance excitation and to internal interactions between modes. Starting from a parametric nonlinear model based on a previously developed Lagrangian formulation, asymptotic treatment of the equations of motion was first performed to characterize the nonlinearity of the lowest nonlinear normal modes of the system. They were found to be characterized by a softening nonlinearity associated with the stiffness terms. Subsequently, a direct time integration of the equations of motion was performed to compute the frequency response curves (FRCs) when the system is subjected to direct harmonic excitations causing the primary resonance of the lowest skew-symmetric mode shape. The method of multiple scales was then employed to study the bifurcation behavior and deliver closed-form expressions of the FRCs and of the loci of the fold bifurcation points, which provide the stability regions of the system. Furthermore, conditions for the onset of internal resonances between the lowest roller battery modes were found, and a 2:1 resonance between the third and first modes of the system was investigated in the case of harmonic excitation having a frequency close to the first mode and the third mode, respectively.

Subharmonic Excitation for Electrically Actuated Microbeams

2005 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Mohammad I. Younis
Keyword(s):  
Frequency Response ◽  
Rf Mems ◽  
Electrostatic Force ◽  
Primary Resonance ◽  
Resonance Excitation ◽  
Global Dynamics ◽  
Response Curves ◽  
Electrically Actuated ◽  
Subharmonic Excitation ◽  
Order Of Magnitude

We present analysis of the global dynamics of electrically actuated microbeams under subharmonic excitation. The microbeams are excited by a DC electrostatic force and an AC harmonic force with a frequency tuned near twice their fundamental natural frequencies. We show that the dynamic pull-in instability can occur in this case for an electric load much lower than that predicted with static analysis and the same order-of-magnitude as that predicted in the case of primary-resonance excitation. We show that, once the subharmonic resonance is activated, all frequency-response curves reach pull-in, regardless of the magnitude of the AC forcing. Our results show a limited influence of the quality factor on the frequency response. This result and the fact that the frequency-response curves have very steep passband-to-stopband transitions make the combination of a DC voltage and a subhormonic of order one-half a promising candidate for designing improved high-sensitive RF MEMS filters.

Bifurcation Control for a Kind of Non-Autonomous System with Time Delay

2010 ◽  
Vol 34-35 ◽  
pp. 1752-1756
Author(s):  
Chang Zhao Qian ◽  
Zhi Wen Wang ◽  
Chuang Wen Dong ◽  
Yang Liu
Keyword(s):  
Time Delay ◽  
Perturbation Method ◽  
Time Delays ◽  
Autonomous System ◽  
Primary Resonance ◽  
Stable Region ◽  
Bifurcation Equation ◽  
Van Der Pol ◽  
Average Equation ◽  
System With Time Delay

A forced van der Pol system with two time-delays is studied. The central aim is analyzing primary resonance of this system. Perturbation method is used to obtain the average equation and bifurcation equation with time-delays. Based on the average equation, the stable region of this system is discussed. Based on the bifurcation equation, the multivalued property of response amplitude is studied. The result indicates that this system can be well controlled with time delays.

scholarly journals Primary Resonance of van der Pol Oscillator under Fractional-Order Delayed Feedback and Forced Excitation

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Jufeng Chen ◽  
Xianghong Li ◽  
Jianhua Tang ◽  
Yafeng Liu
Keyword(s):  
Analytical Solution ◽  
Fractional Order ◽  
Approximate Analytical Solution ◽  
Dynamical Behavior ◽  
Primary Resonance ◽  
Delayed Feedback ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Similar System ◽  
Forced Excitation

The primary resonance of van der Pol oscillator under fractional-order delayed negative feedback and forced excitation is studied. Firstly, the approximate analytical solution is obtained based on the averaging method, and it could be found that the fractional-order delayed feedback has not only the property of delayed velocity feedback but also that of delayed displacement feedback. Moreover, the amplitude-frequency equation for the steady-state solution is established, and its stability conditions are also obtained. Then, the results of the approximate analytical solution and numerical integration are compared and analyzed. The agreement between the two methods is very high, so that the correctness and accuracy of the approximate analytical solution are verified. Finally, the effects of all the parameters in the fractional-order delayed feedback on the amplitude-frequency curves are analyzed. It could be concluded that fractional-order delayed feedback has important influences on the dynamical behavior of van der Pol oscillator, which is very significant to the optimization and control of a similar system.

Effect of Two-to-One Internal Resonances in a Nonlinearly Restrained Fluid Valve

1999 ◽  
Vol 121 (1) ◽  
pp. 59-63 ◽  
Author(s):  
G. Anlas¸
Keyword(s):  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Distributed Parameter ◽  
Parameter System ◽  
Response Curves ◽  
Internal Resonances ◽  
Relief Valve ◽  
Steady State Solutions

The effect of two-to-one internal resonances on the nonlinear response of a pressure relief valve is studied. The fluid valve is modeled as a distributed parameter system at one end and nonlinearly restrained at the other. The method of multiple scales is used to solve the system of partial differential equation and boundary conditions. Frequency-response curves are presented for the primary resonance of either mode in the presence of a two-to-one internal resonance. Stability of the steady-state solutions is investigated. Parameters of the system leading to two-to-one internal resonances are tabulated.

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