Beating Effects in a Single Nonlinear Dynamical System in a Neighborhood of the Resonance

2016 ◽  
Vol 849 ◽  
pp. 76-83
Author(s):  
Jiří Náprstek ◽  
Cyril Fischer
Keyword(s):  
Harmonic Balance ◽  
Excitation Frequency ◽  
Nonlinear Dynamical System ◽  
Numerical Continuation ◽  
System Response ◽  
Algebraic Equations ◽  
Single Degree Of Freedom ◽  
Nonlinear Dynamical ◽  
The Difference ◽  
Eigen Frequency

The exact coincidence of external excitation and basic eigen-frequency of a single degree of freedom (SDOF) nonlinear system produces stationary response with constant amplitude and phase shift. When the excitation frequency differs from the system eigen-frequency, various types of quasi-periodic response occur having a character of a beating process. The period of beating changes from infinity in the resonance point until a couple of excitation periods outside the resonance area. Theabove mentioned phenomena have been identified in many papers including authors’ contributions. Nevertheless, investigation of internal structure of a quasi-period and its dependence on the difference of excitation and eigen-frequency is still missing. Combinations of harmonic balance and small parameter methods are used for qualitative analysis of the system in mono- and multi-harmonic versions. They lead to nonlinear differential and algebraic equations serving as a basis for qualitativeanalytic estimation or numerical description of characteristics of the quasi-periodic system response. Zero, first and second level perturbation techniques are used. Appearance, stability and neighborhood of limit cycles is evaluated. Numerical phases are based on simulation processes and numerical continuation tools. Parametric evaluation and illustrating examples are presented.

Analytical Solutions of a First-Order Quadratic Nonlinear System With Parametrical and Periodical Excitation

2016 ◽  
Author(s):  
Guopeng Zhou ◽  
Albert C. J. Luo ◽  
Naiding Zhou ◽  
Feng Liang
Keyword(s):  
Dynamical System ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Numerical Solutions ◽  
Excitation Frequency ◽  
Nonlinear Dynamical System ◽  
Harmonic Balance Method ◽  
Nonlinear Dynamical ◽  
Approximate Analytical Solutions ◽  
Generalized Harmonic Balance Method

In this paper, a quadratic nonlinear dynamical system with two periodic excitation forces is discussed. Analytical period-1 motions of such dynamical system are obtained by using generalized harmonic balance method. Stability analysis is carried out via eigenvalues analysis. To verify approximate analytical solutions, numerical simulations are completed to compare analytic and numerical solutions of the dynamical system, the approximate precision is guaranteed with appropriate harmonic balance terms. More harmonic terms should be employed to guarantee good approximation of periodic motions if excitation frequency is small. Furthermore, infinite harmonic balance terms must be introduced for chaotic systems.

A New Method for Equilibria and Stability Analysis of Nonlinear Dynamical Systems

2014 ◽  
Vol 534 ◽  
pp. 131-136
Author(s):  
Long Cao ◽  
Yi Hua Cao
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Numerical Continuation ◽  
Vehicle Model ◽  
Nonlinear Dynamical ◽  
Novel Method ◽  
Linearized System ◽  
Continuation Algorithm

A novel method based on numerical continuation algorithm for equilibria and stability analysis of nonlinear dynamical system is introduced and applied to an aircraft vehicle model. Dynamical systems are usually modeled with differential equations, while their equilibria and stability analysis are pure algebraic problems. The newly-proposed method in this paper provides a way to solve the equilibrium equation and the eigenvalues of the locally linearized system simultaneously, which avoids QR iterations and can save much time.

scholarly journals A Sensitivity Analysis of Simulated Infiltration Rates to Uncertain Discretization in the Moisture Content Domain

Water ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 1192
Author(s):  
Lulu Liu ◽  
Han Yu
Keyword(s):  
Moisture Content ◽  
Nonlinear Dynamical System ◽  
Infiltration Rate ◽  
Hydrologic Model ◽  
Water Infiltration ◽  
Nonlinear Dynamical ◽  
Pore Sizes ◽  
Infiltration Rates ◽  
The Difference ◽  
Region Scale

An unconditionally mass conservative hydrologic model proposed by Talbot and Ogden provides an effective and fast technique for estimating region-scale water infiltration. It discretizes soil moisture content into a proper but uncertain number of hydraulically interacting bins such that each bin represents a collection of pore sizes. To simulate rainfall-infiltration, a two-step alternating process runs until completion; and these two steps are surface water infiltration into bins and redistribution of inter-bin flow. Therefore, a nonlinear dynamical system in time is generated based on different bin front depths. In this study, using rigorous mathematical analysis first reveals that more bins can produce larger infiltration fluxes, and the overall flux variation is nonlinear with respect to the number of bins. It significantly implies that a greater variety of pore sizes produces a larger infiltration rate. An asymptotic analysis shows a finite change in infiltration rates for an infinite number of bins, which maximizes the heterogeneity of pore sizes. A corollary proves that the difference in the predicted infiltration rates using this model can be quantitatively bounded under a specific depth ratio of the deepest to the shallowest bin fronts. The theoretical results are demonstrated using numerical experiments in coarse and fine textured soils. Further studies will extend the analysis to the general selection of a suitable number of bins.

An Analysis of Pulse Control for Simple Mechanical Systems

1985 ◽  
Vol 107 (2) ◽  
pp. 123-131 ◽  
Author(s):  
Z. Prucz ◽  
T. T. Soong ◽  
A. Reinhorn
Keyword(s):  
Control Algorithm ◽  
Control Method ◽  
Excitation Frequency ◽  
Mechanical Systems ◽  
System Response ◽  
Control Parameters ◽  
Pulse Control ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
On Line

An efficient pulse control method for insuring safety of simple mechanical systems is developed and its sensitivity to the excitation frequency content and to various control parameters is studied. The control algorithm, consisting of applying pulse forces in a feedback fashion, is designed to insure that maximum system response is limited to safe values at all times. It is shown that the proposed algorithm is simple to implement and is efficient in controlling peak response in terms of on-line computation and pulse energy required. The technique is illustrated and analyzed for a single-degree-of-freedom linear system.

Stochastic Response of Hysteresis System Under Combined Periodic and Stochastic Excitation Via the Statistical Linearization Method

2021 ◽  
Vol 88 (5) ◽  
Author(s):  
Fan Kong ◽  
Pol D. Spanos
Keyword(s):  
Harmonic Balance Method ◽  
Linearization Method ◽  
System Response ◽  
Statistical Linearization ◽  
Algebraic Equations ◽  
Stochastic Response ◽  
Single Degree Of Freedom ◽  
Stochastic Component ◽  
Linear Differential

Abstract A statistical linearization approach is proposed for determining the response of the single-degree-of-freedom of the classical Bouc–Wen hysteretic system subjected to excitation both with harmonic and stochastic components. The method is based on representing the system response as a combination of a harmonic and of a zero-mean stochastic component. Specifically, first, the equation of motion is decomposed into a set of two coupled non-linear differential equations in terms of the unknown deterministic and stochastic response components. Next, the harmonic balance method and the statistical linearization method are used for the determination of the Fourier coefficients of the deterministic component, and the variance of the stochastic component, respectively. This yields a set of coupled algebraic equations which can be solved by any of the standard apropos algorithms. Pertinent numerical examples demonstrate the applicability, and reliability of the proposed method.

scholarly journals Minimal blowing pressure allowing periodic oscillations in a simplified reed musical instrument model: Bouasse-Benade prescription assessed through numerical continuation

Acta Acustica ◽  
2020 ◽  
Vol 4 (6) ◽  
pp. 27
Author(s):  
Joel Gilbert ◽  
Sylvain Maugeais ◽  
Christophe Vergez
Keyword(s):  
Dynamical System ◽  
Continuation Method ◽  
Nonlinear Dynamical System ◽  
Period Doubling ◽  
Acoustic Resonance ◽  
Musical Instrument ◽  
Numerical Continuation ◽  
Physical Parameters ◽  
Periodic Oscillations ◽  
Nonlinear Dynamical

A reed instrument model with N acoustical modes can be described as a 2N dimensional autonomous nonlinear dynamical system. Here, a simplified model of a reed-like instrument having two quasi-harmonic resonances, represented by a four dimensional dynamical system, is studied using the continuation and bifurcation software AUTO. Bifurcation diagrams of equilibria and periodic solutions are explored with respect to the blowing mouth pressure, with focus on amplitude and frequency evolutions along the different solution branches. Equilibria and periodic regimes are connected through Hopf bifurcations, which are found to be direct or inverse depending on the physical parameters values. Emerging periodic regimes mainly supported by either the first acoustic resonance (first register) or the second acoustic resonance (second register) are successfully identified by the model. An additional periodic branch is also found to emerge from the branch of the second register through a period-doubling bifurcation. The evolution of the oscillation frequency along each branch of the periodic regimes is also predicted by the continuation method. Stability along each branch is computed as well. Some of the results are interpreted in terms of the ease of playing of the reed instrument. The effect of the inharmonicity between the first two impedance peaks is observed both when the amplitude of the first is greater than the second, as well as the inverse case. In both cases, the blowing pressure that results in periodic oscillations has a lowest value when the two resonances are harmonic, a theoretical illustration of the Bouasse-Benade prescription.

scholarly journals A nonlinear model of the hand-arm system and parameters identification using vibration transmissibility

2018 ◽  
Vol 241 ◽  
pp. 01014
Author(s):  
Nahid Hida ◽  
Mohamed Abid ◽  
Faouzi Lakrad
Keyword(s):  
Nonlinear Model ◽  
Harmonic Balance ◽  
Grip Force ◽  
Excitation Frequency ◽  
Harmonic Balance Method ◽  
Single Degree Of Freedom ◽  
Identification Process ◽  
Harmonic Vibrations ◽  
Single Degree ◽  
Vibration Transmissibility

In the present paper a lumped single degree-of-freedom nonlinear model is used to study biodynamic responses of the hand arm system (HAS) under harmonic vibrations. Then, the harmonic balance method is implemented to derive the vibration transmissibility. Furthermore,Padé approximations are used in the identification process of biodynamic characteristics of the HAS model. This process is based on minimizing the distance between the theoretical and the experimentally measured transmissibilities. The proposed identification workflow is applied to vibrations at the wrist in two cases: 1) the transmissibility versus the grip force for fixed excitation frequencies, and 2) the transmissibility versus the excitation frequency for fixed grip force.

scholarly journals An ordinary differential equation approach for nonlinear programming and nonlinear complementary problem

2020 ◽  
Vol 1 (1) ◽  
pp. 1
Author(s):  
Irfan Nurhidayat ◽  
Zijun Hao ◽  
Chu-chin Hu ◽  
Jein-Shan Chen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Time Integration ◽  
Nonlinear Dynamical System ◽  
Special Technique ◽  
Algebraic Equations ◽  
Approximation Solution ◽  
Nonlinear Dynamical ◽  
Group Preserving Scheme ◽  
Original Time

We consider an ordinary differential equation (ODE) approach for solving non- linear programming (NLP) and nonlinear complementary problem (NCP). The Karush- Kuhn Tucker (KKT) optimality conditions can be converted to NCP. Based on the Fischer-Burmeister (FB) function and the Natural-Residual (NR) function are obtained the new NCP-functions. A special technique is employed to reformulate of the NCP as the system of nonlinear algebraic equations (NAEs) later on reformulated once more by force. of an original time-like function into an ODE. Afterwards, a group preserving scheme (GPS) is a package to reformulate an ODE into the new numerical equation in a way the ODEs system is designed into a nonlinear dynamical system (NDS) and is continued to a discovery the new numerical equation through activating the Lorentz group SO0(n, 1) and its Lie algebra so(n, 1). Lastly, the fictitious time integration method (FTIM) is utilized into this new numerical equation to determine an approximation solution at the numerical experiments area.

SPATIOTEMPORAL STRUCTURES IN UNDOPED PHOTOEXCITED SEMICONDUCTOR SUPERLATTICES

2001 ◽  
Vol 11 (11) ◽  
pp. 2817-2822 ◽  
Author(s):  
A. PERALES ◽  
L. L. BONILLA ◽  
M. MOSCOSO ◽  
J. GALÁN
Keyword(s):  
Degrees Of Freedom ◽  
Dynamical Behavior ◽  
Nonlinear Dynamical System ◽  
Nonlinear Behavior ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
Semiconductor Superlattices ◽  
Nonlinear Dynamical ◽  
Strongly Nonlinear ◽  
Parameter Ranges

Semiconductor superlattices are a very interesting example of a nonlinear dynamical system with a large number of degrees of freedom. They show a strongly nonlinear behavior and they are well suited for the observation of current instabilities. In the present work, the dynamical behavior of undoped photoexcited superlattices has been analyzed by numerical continuation methods and bifurcation theory within the framework of a simple drift-diffusion model. The control parameters are the applied dc voltage and the carrier density, which are related to the laser power. We compile our results in a phase diagram and locate the lines where the system undergoes qualitative changes of behavior. The oscillatory regions are related to the appearance and disappearance of Hopf tongue crossings, for which oscillations appear as sub- or supercritical bifurcations. This implies the existence of voltage windows of current oscillations and hysteresis in appropriate parameter ranges, which agrees with recent experimental observations.

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