scholarly journals Analytical Solutions of Some Strong Nonlinear Oscillators

2021 ◽  
Author(s):  
Alvaro Humberto Salas ◽  
Samir Abd El-Hakim El-Tantawy
Keyword(s):  
Analytical Solutions ◽  
Initial Conditions ◽  
Exact Analytical Solution ◽  
Main Idea ◽  
Analytical Techniques ◽  
Elliptic Functions ◽  
Nonlinear Oscillators ◽  
Continuous Functions ◽  
Soliton Theory ◽  
Quintic Polynomial

Oscillators are omnipresent; most of them are inherently nonlinear. Though a nonlinear equation mostly does not yield an exact analytic solution for itself, plethora of elementary yet practical techniques exist for extracting important information about the solution of equation. The purpose of this chapter is to introduce some new techniques for the readers which are carefully illustrated using mainly the examples of Duffing’s oscillator. Using the exact analytical solution to cubic Duffing and cubic-quinbic Duffing oscillators, we describe the way other conservative and some non conservative damped nonlinear oscillators may be studied using analytical techniques described here. We do not make use of perturbation techniques. However, some comparison with such methods are performed. We consider oscillators having the form x¨+fx=0 as well as x¨+2εẋ+fx=Ft, where x=xt and f=fx and Ft are continuous functions. In the present chapter, sometimes we will use f−x=−fx and take the approximation fx≈∑j=1Npjxj, where j=1,3,5,⋯N only odd integer values and x∈−AA. Moreover, we will take the approximation fx≈∑j=0Npjxj, where j=1,2,3,⋯N, and x∈−AA. Arbitrary initial conditions are considered. The main idea is to approximate the function f=fx by means of some suitable cubic or quintic polynomial. The analytical solutions are expressed in terms of the Jacobian and Weierstrass elliptic functions. Applications to plasma physics, electronic circuits, soliton theory, and engineering are provided.

scholarly journals Nonlinear Behavior of a Magnetic Bearing System

1995 ◽  
Vol 117 (3) ◽  
pp. 582-588 ◽  
Author(s):  
L. N. Virgin ◽  
T. F. Walsh ◽  
J. D. Knight
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Nonlinear Behavior ◽  
Analytical Techniques ◽  
Magnetic Bearing ◽  
Forced Response ◽  
The Mathematical Model ◽  
Two Degree Of Freedom ◽  
Sensitivity To Initial Conditions ◽  
Bearing System

This paper describes the results of a study into the dynamic behavior of a magnetic bearing system. The research focuses attention on the influence of nonlinearities on the forced response of a two-degree-of-freedom rotating mass suspended by magnetic bearings and subject to rotating unbalance and feedback control. Geometric coupling between the degrees of freedom leads to a pair of nonlinear ordinary differential equations, which are then solved using both numerical simulation and approximate analytical techniques. The system exhibits a variety of interesting and somewhat unexpected phenomena including various amplitude driven bifurcational events, sensitivity to initial conditions, and the complete loss of stability associated with the escape from the potential well in which the system can be thought to be oscillating. An approximate criterion to avoid this last possibility is developed based on concepts of limiting the response of the system. The present paper may be considered as an extension to an earlier study by the same authors, which described the practical context of the work, free vibration, control aspects, and derivation of the mathematical model.

scholarly journals APPLICATION OF ABOODH TRANSFORM ON SOME FRACTIONAL ORDER MATHEMATICAL MODELS

2020 ◽  
Vol 20 (3) ◽  
pp. 661-672
Author(s):  
JAWARIA TARIQ ◽  
JAMSHAD AHMAD
Keyword(s):  
Mathematical Models ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Iteration Method ◽  
Analytical Techniques ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Physical Phenomena ◽  
Convergent Solution

In this work, a new emerging analytical techniques variational iteration method combine with Aboodh transform has been applied to find out the significant important analytical and convergent solution of some mathematical models of fractional order. These mathematical models are of great interest in engineering and physics. The derivative is in Caputo’s sense. These analytical solutions are continuous that can be used to understand the physical phenomena without taking interpolation concept. The obtained solutions indicate the validity and great potential of Aboodh transform with the variational iteration method and show that the proposed method is a good scheme. Graphically, the movements of some solutions are presented at different values of fractional order.

scholarly journals PARALLEL COMPUTATIONS IN STUDIES OF DEPENDENCE OF GAS DYNAMIC PARAMETERS OF UPWARD SWIRLING FLOW OF GAS ON BLOWING VELOCITY

2016 ◽  
pp. 92-97
Author(s):  
R. E. Volkov ◽  
A. G. Obukhov
Keyword(s):  
Stokes Equations ◽  
Swirling Flow ◽  
Initial Conditions ◽  
Exact Analytical Solution ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Heat Conducting ◽  
Computational Domain ◽  
Navier Stokes Equations ◽  
Gas Dynamic

The rectangular parallelepiped explicit difference schemes for the numerical solution of the complete built system of Navier-Stokes equations. These solutions describe the three-dimensional flow of a compressible viscous heat-conducting gas in a rising swirling flows, provided the forces of gravity and Coriolis. This assumes constancy of the coefficient of viscosity and thermal conductivity. The initial conditions are the features that are the exact analytical solution of the complete Navier-Stokes equations. Propose specific boundary conditions under which the upward flow of gas is modeled by blowing through the square hole in the upper surface of the computational domain. A variant of parallelization algorithm for calculating gas dynamic and energy characteristics. The results of calculations of gasdynamic parameters dependency on the speed of the vertical blowing by the time the flow of a steady state flow.

scholarly journals Mandelbrot, Fama and the emergence of econophysics

2016 ◽  
Vol 35 (69) ◽  
pp. 637-662
Author(s):  
Boris Salazar Trujillo
Keyword(s):  
Analytical Solutions ◽  
Financial Economics ◽  
Analytical Techniques ◽  
Infinite Variance ◽  
Pareto Distributions

It is argued that Mandelbrot's stable Lévy-Pareto distributions were not accepted into the emerging field of financial economics due to their incompatibility with the analytical techniques and properties of equilibrium economics, and to the absence -both in physics and in economics- of analytical solutions to the infinite variance associated with those distributions. Whilst physicists made stable Lévy distributions plausible, creating Econophysics in the meantime, economists just forgot about them, suggesting their strong bias towards desirable properties and against established facts.

scholarly journals Improving Student Learning Outcomes Through Story Card Media to Find The Main Idea of a Paragraph in Class VI SDN Luwunggede 04

2020 ◽  
Vol 3 (4) ◽  
pp. 640
Author(s):  
Kirno Kirno
Keyword(s):  
Action Research ◽  
Student Learning ◽  
Learning Outcomes ◽  
Initial Conditions ◽  
Main Idea ◽  
Student Learning Outcomes ◽  
Final Conclusion ◽  
Classroom Action Research ◽  
Improving Student Learning ◽  
Improve Student Learning

<p><em>The purpose of this study is to improve student learning outcomes in Indonesian lessons in elementary school grade VI by using story cards to find the main idea of a paragraph. The research conducted was Classroom Action Research (CAR) in two cycles, with each cycle having one meeting. The stages of each cycle are the stages of planning, implementation, observation and reflection. Data collection techniques are observation and tests. Based on data analysis, it was found that the use of story card media to find the main idea of paragraphs in Indonesian language lessons to determine the main idea can improve student learning outcomes as indicated by the level of mastery learning in the initial conditions of 40% to 65% in the first cycle, and increased to 80% in cycle II. The final conclusion from the implementation of this classroom action research is that the use of story card media to find the main idea of a paragraph is able to improve student learning outcomes in Indonesian language lessons to determine the main idea at SD Negeri Luwunggede 04, Larangan, Brebes.</em></p>

Identification of Nonlinear Systems Parameters Using Polyspectral Measurements and Analysis

1995 ◽  
Author(s):  
Muhammad R. Hajj ◽  
Ali H. Nayfeh ◽  
Pavol Popovic
Keyword(s):  
Experimental Study ◽  
Nonlinear Systems ◽  
Analytical Solutions ◽  
Analytical Techniques ◽  
Modal Interactions ◽  
Unknown Parameters ◽  
Nonlinear Parameters ◽  
The Subject ◽  
Beam Frame ◽  
Subharmonic Resonances

Abstract Experimental and analytical techniques that characterize nonlinear modal interactions in structures are used to quantify parameters in representative nonlinear systems. The subject of the experimental study is a three-beam frame. Subharmonic resonances and interaction between widely spaced modes are exploited to determine nonlinear parameters in models that represent these interactions. The phases of the auto-bispectra of the response of this structure appear in the analytical solutions of the representative models. Values of these phases could thus aid in determining other unknown parameters of nonlinear systems.

scholarly journals He's frequency–amplitude formulation for nonlinear oscillators using Jacobi elliptic functions

2020 ◽  
Vol 39 (4) ◽  
pp. 1216-1223 ◽  
Author(s):  
Alex Elías-Zúñiga ◽  
Luis Manuel Palacios-Pineda ◽  
Isaac H Jiménez-Cedeño ◽  
Oscar Martínez-Romero ◽  
Daniel Olvera Trejo
Keyword(s):  
Elliptic Functions ◽  
Nonlinear Oscillators ◽  
Numerical Results ◽  
Jacobi Elliptic Functions ◽  
Analytical Expression ◽  
Analytical Frequency

In this work, the Duffing’s type analytical frequency–amplitude relationship for nonlinear oscillators is derived by using Hés formulation and Jacobi elliptic functions. Comparison of the numerical results obtained from the derived analytical expression using Jacobi elliptic functions with respect to the exact ones is performed by considering weak and strong Duffing’s nonlinear oscillators.

Improved Finite Elements for Vibration Analysis of Tapered Beams

1973 ◽  
Vol 24 (1) ◽  
pp. 39-46 ◽  
Author(s):  
J Thomas ◽  
E Dokumaci
Keyword(s):  
Finite Elements ◽  
Analytical Solutions ◽  
Vibration Analysis ◽  
Basic Element ◽  
Quintic Polynomial ◽  
End Conditions

SummaryThis paper gives two improved tapered elements for vibration analysis derived using quintic polynomial displacement functions. The elements employ different nodal continuity conditions. Results are compared with those given by the basic cubic approximations and analytical solutions for various end conditions. The advantages of the new elements over the basic element are discussed.

Analytical solutions and expressions of the propagator for Bloch equations

2020 ◽  
Vol 34 (18) ◽  
pp. 2050158
Author(s):  
Heung-Ryoul Noh
Keyword(s):  
Analytical Solutions ◽  
Initial Conditions ◽  
Secular Equation ◽  
Magnetization Vector ◽  
Bloch Equations ◽  
Optical Bloch Equations ◽  
Parameter Spaces ◽  
Pulse Spectroscopy ◽  
Analytical Expressions ◽  
Composite Laser

In this paper, we present analytical solutions to the Bloch equations. After solving the secular equation for the eigenvalues, derived from the Bloch equations, analytical solutions for the temporal evolution of the magnetization vector are obtained at arbitrary initial conditions. Subsequently, explicit analytical expressions of the propagator for the Bloch equations and optical Bloch equations are obtained. Compared to the results of existing analytical studies, the present results are more succinct and rigorous, and they can predict the behavior of the propagator in different regions of parameter spaces. The analytical solutions to the propagator can be directly used in composite laser-pulse spectroscopy.

scholarly journals The Analytical Analysis of Time-Fractional Fornberg–Whitham Equations

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 987 ◽  
Author(s):  
A. A. Alderremy ◽  
Hassan Khan ◽  
Rasool Shah ◽  
Shaban Aly ◽  
Dumitru Baleanu
Keyword(s):  
Mathematical Modeling ◽  
Analytical Solution ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Decomposition Method ◽  
Analytical Techniques ◽  
Whitham Equations ◽  
Suggested Technique ◽  
Approximate Analytical Solutions ◽  
Physical Phenomena

This article is dealing with the analytical solution of Fornberg–Whitham equations in fractional view of Caputo operator. The effective method among the analytical techniques, natural transform decomposition method, is implemented to handle the solutions of the proposed problems. The approximate analytical solutions of nonlinear numerical problems are determined to confirm the validity of the suggested technique. The solution of the fractional-order problems are investigated for the suggested mathematical models. The solutions-graphs are then plotted to understand the effectiveness of fractional-order mathematical modeling over integer-order modeling. It is observed that the derived solutions have a closed resemblance with the actual solutions. Moreover, using fractional-order modeling various dynamics can be analyzed which can provide sophisticated information about physical phenomena. The simple and straight-forward procedure of the suggested technique is the preferable point and thus can be used to solve other nonlinear fractional problems.

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