scholarly journals Perturbation solutions of fifth order oscillatory nonlinear systems

2011 ◽  
Vol 16 (2) ◽  
pp. 123-134
Author(s):  
M. Ali Akbar ◽  
Sk. Tanzer Ahmed Siddique
Keyword(s):  
Degrees Of Freedom ◽  
Nonlinear Differential Equations ◽  
Weakly Nonlinear ◽  
Physical Systems ◽  
Oscillatory Systems ◽  
Engineering Problems ◽  
Kbm Method ◽  
Fifth Order ◽  
Method Show ◽  
Good Agreement

Oscillatory systems play an important role in the nature. Many engineering problems and physical systems of fifth degrees of freedom are oscillatory and their governing equations are fifth order nonlinear differential equations. To investigate the solution of fifth order weakly nonlinear oscillatory systems, in this article the Krylov–Bogoliubov–Mitropolskii (KBM) method has been extended and desired solution is found. An example is solved to illustrate the method. The results obtain by the extended KBM method show good agreement with those obtained by numerical method.

Asymptotic Solutions of Fifth Order Overdamped-Oscillatory Nonlinear Systems

2020 ◽  
Author(s):  
Harun-Or-Roshid ◽  
M. Zulfikar Ali
Keyword(s):  
Numerical Solutions ◽  
Oscillatory System ◽  
Kutta Method ◽  
Natural Phenomena ◽  
Extended Technique ◽  
Runge Kutta Method ◽  
Engineering Problems ◽  
Kbm Method ◽  
Fifth Order ◽  
Good Agreement

Combo overdamp-oscillatory system plays an important role in natural phenomena in many engineering problems. In this paper, fifth order nonlinear damped-oscillatory differential system is studied to investigate an asymptotic analytical approximate solution in the fashion of overdamp-oscillations via an extension of the Krylov-Bogoliubov-Mitropolskii (KBM) method. The proposed method is demonstrated by its applications on a Duffing oscillators in the combined form of overdamp and oscillatory effects. The result obtained by the presented extended technique good agreement with the numerical solutions of the fourth order Runge-Kutta method.

scholarly journals Perturbation Solutions to Fifth Order Over-damped Nonlinear Systems

2019 ◽  
pp. 1-11
Author(s):  
Harun-Or- Roshid ◽  
M. Zulfikar Ali ◽  
Pinakee Dey ◽  
M. Ali Akbar
Keyword(s):  
Nonlinear Systems ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
First Order ◽  
Engineering Problems ◽  
Kbm Method ◽  
Fifth Order ◽  
Physical Phenomena ◽  
Good Agreement ◽  
Perturbation Solutions

Fifth order over-damp nonlinear differential systems can be used to describe many engineering problems and physical phenomena occur in the nature. In this article, the Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to investigate the solution of a certain fifth order over-damp nonlinear systems and desired result has been found. The implementation of the presented method is illustrated by an example. The first order analytical approximate solutions obtained by the method for different initial conditions show a good agreement with those obtains by numerical method.

ON THE GLOBAL BEHAVIOR OF SELF-OSCILLATORY SYSTEMS

1993 ◽  
Vol 03 (01) ◽  
pp. 195-200 ◽  
Author(s):  
A.D. MOROZOV
Keyword(s):  
Hamiltonian System ◽  
Invariant Torus ◽  
Degrees Of Freedom ◽  
Systems Approach ◽  
Global Behavior ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
System Parameters ◽  
Oscillatory Systems ◽  
Good Agreement

The problem of the global behavior of the system with 3/2 degrees of freedom is considered in this paper. The results of the theoretical investigation of the case when the systems approach a nonlinear Hamiltonian system are presented. Using, for example, the Duffing-van der Pol equation, computer analysis gives good agreement with theory when the external force has moderate amplitude values (“miracle of a small parameter”). The bifurcations of the invariant torus and resonances are analyzed as a function of the system parameters.

scholarly journals Weakly nonlinear analysis of thermoacoustic bifurcations in the Rijke tube

2016 ◽  
Vol 805 ◽  
pp. 523-550 ◽  
Author(s):  
Alessandro Orchini ◽  
Georgios Rigas ◽  
Matthew P. Juniper
Keyword(s):  
Dynamical Evolution ◽  
Hopf Bifurcations ◽  
Solvability Conditions ◽  
Rijke Tube ◽  
Nonlinear Dynamical ◽  
Weakly Nonlinear ◽  
The Cost ◽  
Fifth Order ◽  
Thermoacoustic Oscillations ◽  
Good Agreement

In this study we present a theoretical weakly nonlinear framework for the prediction of thermoacoustic oscillations close to Hopf bifurcations. We demonstrate the method for a thermoacoustic network that describes the dynamics of an electrically heated Rijke tube. We solve the weakly nonlinear equations order by order, discuss their contribution on the overall dynamics and show how solvability conditions at odd orders give rise to Stuart–Landau equations. These equations, combined together, describe the nonlinear dynamical evolution of the oscillations’ amplitude and their frequency. Because we retain the contribution of several acoustic modes in the thermoacoustic system, the use of adjoint methods is required to derive the Landau coefficients. The analysis is performed up to fifth order and compared with time domain simulations, showing good agreement. The theoretical framework presented here can be used to reduce the cost of investigating oscillations and subcritical phenomena close to Hopf bifurcations in numerical simulations and experiments and can be readily extended to consider, e.g. the weakly nonlinear interaction of two unstable thermoacoustic modes.

scholarly journals A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 920
Author(s):  
Chukwuma Ogbonnaya ◽  
Chamil Abeykoon ◽  
Adel Nasser ◽  
Ali Turan
Keyword(s):  
Numerical Models ◽  
Problem Formulation ◽  
Science And Engineering ◽  
Physical Systems ◽  
Engineering Problems ◽  
Parametric Studies ◽  
Independent Variable ◽  
Transcendental Equations ◽  
The Waves ◽  
Modelling Approach

A system of transcendental equations (SoTE) is a set of simultaneous equations containing at least a transcendental function. Solutions involving transcendental equations are often problematic, particularly in the form of a system of equations. This challenge has limited the number of equations, with inter-related multi-functions and multi-variables, often included in the mathematical modelling of physical systems during problem formulation. Here, we presented detailed steps for using a code-based modelling approach for solving SoTEs that may be encountered in science and engineering problems. A SoTE comprising six functions, including Sine-Gordon wave functions, was used to illustrate the steps. Parametric studies were performed to visualize how a change in the variables affected the superposition of the waves as the independent variable varies from x1 = 1:0.0005:100 to x1 = 1:5:100. The application of the proposed approach in modelling and simulation of photovoltaic and thermophotovoltaic systems were also highlighted. Overall, solutions to SoTEs present new opportunities for including more functions and variables in numerical models of systems, which will ultimately lead to a more robust representation of physical systems.

scholarly journals Exact mapping between a laser network loss rate and the classical XY Hamiltonian by laser loss control

Nanophotonics ◽  
2020 ◽  
Vol 9 (13) ◽  
pp. 4117-4126 ◽  
Author(s):  
Igor Gershenzon ◽  
Geva Arwas ◽  
Sagie Gadasi ◽  
Chene Tradonsky ◽  
Asher Friesem ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Loss Rate ◽  
Oscillation Amplitude ◽  
Xy Model ◽  
Spin Hamiltonian ◽  
Laser Oscillator ◽  
Physical Systems ◽  
Network State ◽  
Spin Hamiltonians ◽  
Loss Control

AbstractRecently, there has been growing interest in the utilization of physical systems as heuristic optimizers for classical spin Hamiltonians. A prominent approach employs gain-dissipative optical oscillator networks for this purpose. Unfortunately, these systems inherently suffer from an inexact mapping between the oscillator network loss rate and the spin Hamiltonian due to additional degrees of freedom present in the system such as oscillation amplitude. In this work, we theoretically analyze and experimentally demonstrate a scheme for the alleviation of this difficulty. The scheme involves control over the laser oscillator amplitude through modification of individual laser oscillator loss. We demonstrate this approach in a laser network classical XY model simulator based on a digital degenerate cavity laser. We prove that for each XY model energy minimum there corresponds a unique set of laser loss values that leads to a network state with identical oscillation amplitudes and to phase values that coincide with the XY model minimum. We experimentally demonstrate an eight fold improvement in the deviation from the minimal XY energy by employing our proposed solution scheme.

Estimation of the Thermodynamic Properties of Branched Hydrocarbons

2000 ◽  
Vol 122 (3) ◽  
pp. 147-152 ◽  
Author(s):  
Hui He ◽  
Mohamad Metghalchi ◽  
James C. Keck
Keyword(s):  
Thermodynamic Properties ◽  
Degrees Of Freedom ◽  
Statistical Thermodynamic ◽  
Motion Modes ◽  
Branched Alkanes ◽  
Branched Hydrocarbons ◽  
Wide Range ◽  
On Line ◽  
Kinetic Calculations ◽  
Good Agreement

A simple model has been developed to estimate the sensible thermodynamic properties such as Gibbs free energy, enthalpy, heat capacity, and entropy of hydrocarbons over a wide range of temperatures with special attention to the branched molecules. The model is based on statistical thermodynamic expressions incorporating translational, rotational and vibrational motions of the atoms. A method to determine the number of degrees of freedom for different motion modes (bending and torsion) has been established. Branched rotational groups, such as CH3 and OH, have been considered. A modification of the characteristic temperatures for different motion mode has been made which improves the agreement with the exact values for simple cases. The properties of branched alkanes up to 2,3,4,-trimthylpentane have been calculated and the results are in good agreement with the experimental data. A relatively small number of parameters are needed in this model to estimate the sensible thermodynamic properties of a wide range of species. The model may also be used to estimate the properties of molecules and their isomers, which have not been measured, and is simple enough to be easily programmed as a subroutine for on-line kinetic calculations. [S0195-0738(00)00902-X]

Computation and cognition: issues in the foundations of cognitive science

1980 ◽  
Vol 3 (1) ◽  
pp. 111-132 ◽  
Author(s):  
Zenon W. Pylyshyn
Keyword(s):  
Tacit Knowledge ◽  
Degrees Of Freedom ◽  
Ad Hoc ◽  
Physical Systems ◽  
Symbolic Representations ◽  
Natural Domain ◽  
Human Functioning ◽  
Biological Laws ◽  
Post Hoc ◽  
Level Of Analysis

AbstractThe computational view of mind rests on certain intuitions regarding the fundamental similarity between computation and cognition. We examine some of these intuitions and suggest that they derive from the fact that computers and human organisms are both physical systems whose behavior is correctly described as being governed by rules acting on symbolic representations. Some of the implications of this view are discussed. It is suggested that a fundamental hypothesis of this approach (the “proprietary vocabulary hypothesis”) is that there is a natural domain of human functioning (roughly what we intuitively associate with perceiving, reasoning, and acting) that can be addressed exclusively in terms of a formal symbolic or algorithmic vocabulary or level of analysis.Much of the paper elaborates various conditions that need to be met if a literal view of mental activity as computation is to serve as the basis for explanatory theories. The coherence of such a view depends on there being a principled distinction between functions whose explanation requires that we posit internal representations and those that we can appropriately describe as merely instantiating causal physical or biological laws. In this paper the distinction is empirically grounded in a methodological criterion called the “cognitive impenetrability condition.” Functions are said to be cognitively impenetrable if they cannot be influenced by such purely cognitive factors as goals, beliefs, inferences, tacit knowledge, and so on. Such a criterion makes it possible to empirically separate the fixed capacities of mind (called its “functional architecture”) from the particular representations and algorithms used on specific occasions. In order for computational theories to avoid being ad hoc, they must deal effectively with the “degrees of freedom” problem by constraining the extent to which they can be arbitrarily adjusted post hoc to fit some particular set of observations. This in turn requires that the fixed architectural function and the algorithms be independently validated. It is argued that the architectural assumptions implicit in many contemporary models run afoul of the cognitive impenetrability condition, since the required fixed functions are demonstrably sensitive to tacit knowledge and goals. The paper concludes with some tactical suggestions for the development of computational cognitive theories.

Estimating Critical Speeds of Stepped Shafts with Flexible Bearings

1971 ◽  
Vol 8 (03) ◽  
pp. 327-333
Author(s):  
R. H. Salzman
Keyword(s):  
High Speed ◽  
Critical Speed ◽  
Design Stage ◽  
Normal Range ◽  
Critical Speeds ◽  
Stepped Shaft ◽  
Bearing Stiffness ◽  
Variable Support ◽  
Method Show ◽  
Good Agreement

This paper presents a semi-graphical approach for finding the first critical speed of a stepped shaft with finite bearing stiffness. The method is particularly applicable to high-speed turbine rotors with journal bearings. Using Rayleigh's Method and the exact solution for whirling of a uniform shaft with variable support stiffness, estimates of the lowest critical speed are easily obtained which are useful in the design stage. First critical speeds determined by this method show good agreement with values computed by the Prohl Method for the normal range of bearing stiffness. A criterion is also established for determining if the criticals are "bearing critical speeds" or "bending critical speeds," which is of importance in design. Discusser E. G. Baker

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