Research for Coupled van der Pol Systems with Parametric Excitation and Its Application

2017 ◽  
Vol 72 (11) ◽  
pp. 1009-1020 ◽  
Author(s):  
Y. H. Qian ◽  
H. X. Fu
Keyword(s):  
Periodic Solutions ◽  
Parametric Excitation ◽  
Multiple Scales ◽  
Coupled System ◽  
Response Curves ◽  
Internal Resonances ◽  
Van Der Pol ◽  
Nonlinear Dynamic Response ◽  
Coordinate Form ◽  
Homotopy Analysis

AbstractIn this article, we study the primary resonances of van der Pol systems with parametric excitation using the multiple scales method (MSM) and the homotopy analysis method (HAM). First, we study the nonlinear dynamic response of a coupled system with parametric excitation when the ratio of internal resonances are different, and obtain the four-dimensional average equation of the rectangular coordinate form using the MSM, thereby periodic motions are found in the system. Second, using the HAM, we obtain the four periodic solutions, in which there are two sets of in-phase periodic solutions and two sets of out-of-phase periodic solutions. Finally, we obtain the frequency response curves using the MSM and the HAM, in which it is found that the differences could be ignored.

Nonlinear Transverse Vibrations and 3:1 Internal Resonances of a Beam With Multiple Supports

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
E. Özkaya ◽  
S. M. Bağdatlı ◽  
H. R. Öz
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Mode Shapes ◽  
Response Curves ◽  
Internal Resonances ◽  
Transverse Vibrations ◽  
Second Mode ◽  
Euler Bernoulli Beam ◽  
Correction Terms

In this study, nonlinear transverse vibrations of an Euler–Bernoulli beam with multiple supports are considered. The beam is supported with immovable ends. The immovable end conditions cause stretching of neutral axis and introduce cubic nonlinear terms to the equations of motion. Forcing and damping effects are included in the problem. The general arbitrary number of support case is considered at first, and then 3-, 4-, and 5-support cases are investigated. The method of multiple scales is directly applied to the partial differential equations. Natural frequencies and mode shapes for the linear problem are found. The correction terms are obtained from the last order of expansion. Nonlinear frequencies are calculated and then amplitude and phase modulation figures are presented for different forcing and damping cases. The 3:1 internal resonances are investigated. External excitation frequency is applied to the first mode and responses are calculated for the first or second mode. Frequency-response and force-response curves are drawn.

Two timescale harmonic balance. I. Application to autonomous one-dimensional nonlinear oscillators

1992 ◽  
Vol 340 (1659) ◽  
pp. 473-501 ◽  
Keyword(s):  
Periodic Solutions ◽  
Harmonic Balance ◽  
Evolution Equations ◽  
Multiple Scales ◽  
Phase Locking ◽  
Period Doubling ◽  
Nonlinear Oscillators ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Wide Range

Two timescale harmonic balance is a semi-analytical/numerical method for deriving periodic solutions and their stability to a class of nonlinear autonomous and forced oscillator equations of the form ẍ + x = f(x,ẋ,λ) and ẍ + x = f(x,ẋ,λ,t) , where λ is a control parameter. The method incorporates salient features from both the method of harmonic balance and multiple scales, and yet does not require an explicit small parameter. Essentially periodic solutions are formally derived on the basis of a single assumption: ‘that an N harmonic, truncated, Fourier series and its first two derivatives can represent x(t) , ẋ(t) and ẍ(t) respectively’. By seeking x(t) as a series of superharmonics, subharmonics, and ultrasubharmonics it is found that the method works over a wide range of parameter space provided the above assumption holds which, in practice, imposes some ‘problem dependent’ restriction on the magnitude of the nonlinearities. Two timescales, associated with the amplitude and phase variations respectively, are introduced by means of an implicit parameter Є . These timescales permit the construction of a set of amplitude evolution equations together with a corresponding stability criterion. In Part I the method is formulated and applied to three autonomous equations, the van der Pol equation, the modified van der Pol equation, and the van der Pol equation with escape. In this case an expansion in superharmonics is sufficient to reveal Hopf, saddle node and homoclinic bifurcations which are compared with results obtained by numerical integration of the equations. In Part II the method is applied to forced nonlinear oscillators in which the solution for x(t) includes superharmonics, subharmonics, and ultrasubharmonics. The features of period doubling, symmetry breaking, phase locking and the Feigenbaum transition to chaos are examined.

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 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.

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.

scholarly journals Modeling and 1 : 1 Internal Resonance Analysis of Cable-Stayed Shallow Arches

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Jiangen Lv ◽  
Zhicheng Yang ◽  
Xuebin Chen ◽  
Quanke Wu ◽  
Xiaoxia Zeng
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Period Doubling ◽  
Dynamic Responses ◽  
Shallow Arch ◽  
Response Curves ◽  
Internal Resonances ◽  
Essential Information ◽  
Modulation Equations ◽  
Inclined Angle

In this paper, an analytical model of a cable-stayed shallow arch is developed in order to investigate the 1 : 1 internal resonance between modes of a cable and a shallow arch. Integrodifferential equations with quadratic and cubic nonlinearities are used to model the in-plane motion of a simple cable-stayed shallow arch. Nonlinear dynamic responses of a cable-stayed shallow arch subjected to external excitations with simultaneous 1 : 1 internal resonances are investigated. Firstly, the Galerkin method is used to discretize the governing nonlinear integral-partial-differential equations. Secondly, the multiple scales method (MSM) is used to derive the modulation equations of the system under external excitation of the shallow arch. Thirdly, the equilibrium, the periodic, and the chaotic solutions of the modulation equations are also analyzed in detail. The frequency- and force-response curves are obtained by using the Newton–Raphson method in conjunction with the pseudoarclength path-following algorithm. The cascades of period-doubling bifurcations leading to chaos are obtained by applying numerical simulations. Finally, the effects of key parameters on the responses are examined, such as initial tension, inclined angle of the cable, and rise and inclined angle of shallow arch. The comprehensive numerical results and research findings will provide essential information for the safety evaluation of cable-supported structures that have widely been used in civil engineering.

scholarly journals THE HOMOTOPY ANALYSIS METHOD IN BIFURCATION ANALYSIS OF DELAY DIFFERENTIAL EQUATIONS

2012 ◽  
Vol 22 (08) ◽  
pp. 1230024 ◽  
Author(s):  
ANDREA BEL ◽  
WALTER REARTES
Keyword(s):  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Homotopy Analysis Method ◽  
Delay Differential Equations ◽  
Analysis Method ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Homotopy Analysis ◽  
Delay Differential ◽  
Bifurcation And Stability

In this paper we apply the homotopy analysis method (HAM) to study the van der Pol equation with a linear delayed feedback. The paper focuses on the calculation of periodic solutions and associated bifurcations, Hopf, double Hopf, Neimark–Sacker, etc. In particular, we discuss the behavior of the systems in the neighborhoods of double Hopf points. We demonstrate the applicability of HAM to the analysis of bifurcation and stability.

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.

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