In addition, secondary resonance corresponding to strong hard excitation is also discussed. Solutions to the oscillator equation can exhibit extreme nonlinear dynamics, including limit cycles, strange attractors, and. In this paper, we describe the application of the elliptic balance method ebm to obtain a general solution of the forced, damped duffing equation by assuming that the modulus of the jacobian elliptic functions are slowly varying as a function of time. The duffing equation provides a reference text for postgraduate and students and researchers of mechanical engineering and vibration nonlinear dynamics as well as a. Forced harmonic vibration of a duffing oscillator with.
The duffing oscillator moves in a double well potential, sometimes characterized as nonlinear elasticity, with sinusoidal external forcing. Forced duffing oscillator file exchange matlab central. Lecture notes on nonlinear vibrations cornell university. Some numerical methods for obtaining harmomc duffings equation. It first became popular for studying anharmonic oscillations and, later, chaotic nonlinear dynamics in the wake of early studies by the engineer georg duffing 1.
Attractors in duffing equation mathematics stack exchange. The method is based on the approximation by the truncated taylor series about center zero. Discretization of forced duffing system with fractionalorder. The duffing map also called as holmes map is a discretetime dynamical system. Experiments with the duffing oscillator from nonlinear dynamics and chaos by j. In this paper, we use the modified differential transform method to obtain the approximate solution of a nonlinear duffing oscillator with a damping effect under different initial conditions. The duffing map takes a point x n, y n in the plane and maps it to a new point given by. Amplitudefrequency relationship to a fractional duffing oscillator. Numerical solution of duffing equation by using an.
In this paper we are interested in studying the effect of the fractionalorder damping in the forced duffing oscillator before and after applying a discretization process to it. Pdf on jan 10, 2017, j sunday and others published the duffing oscillator. It can travel, for example, from some a place in the pacific to some a coast. University of novi sad, faculty of technical sciences, serbia. The duffing oscillator is a common model for nonlinear phenomena in science and engineering. When the periodic force that drives the system is large, chaotic behavior emerges and the phase space diagram is a strange attractor. New in mathematica 9 parametric differential equations sensitivity of the duffing equation explore the sensitivity of the duffing equation to initial conditions. This is a matlab gui project,very simple for draw a piecewise function.
Approximate solution for the duffingharmonic oscillator. Duffing equation which called japanese attractor or ueda attractor. We use mathematica to find the analytical solutions of the duffing problem without friction. In a mechanical oscillator, m is the system mass, c is the viscous damping coefficient, k and h are stiffness coefficients and f t is the force applied to the system. In 1918, georg duffing introduced a nonlinear oscillator model with a cubic stiffness term in the standard differential equation of an idealized linear harmonic oscillator to describe the hardening spring effect observed in many engineering and physical problems cf. In i kovacic and m j brennan eds, the duffing equation. Approximate solution for the duffingharmonic oscillator by. It describes the forced oscillations of several mechanical and electrical systems e. The oscillator can be used to model rotor subjected to nonlinear damping and stiffness with external excitation. This content was uploaded by our users and we assume good faith they have the permission to share this book. The duffing oscillator is one of the prototype systems of nonlinear dynamics. However, there are simply to many extensions to mention them all.
To simplify the computations that follow, use parametricndsolvevalue with only the initial conditions as parameters and other values fixed. Experiments with duffing oscillator ucsd mathematics. Duffings equation an overview sciencedirect topics. I dont see how you are passing the values of the parameters e. Amplitudefrequency relationship to a fractional duffing oscillator arising in microphysics and. Mathematica could be used to bring students to use jacobi elliptic functions without effort. We have suggested a numerical approach, which is based on an improved taylor matrix method, for solving duffing differential equations. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients. It is an example of a dynamical system that exhibits chaotic behavior. Each chapter is written by an expert contributor in. Exact solutions to cubic duffing equation by leaf functions.
Analogue electrical circuit for simulation of the duffing. Forced harmonic vibration of a duffing oscillator with linear. If tr 0, then both eigenvalues are positive and the solution becomes unbounded as t goes to in. The forced duffing oscillator exhibits behavior ranging from limit cycles to chaos due to its nonlinear dynamics. From this solution, the maximum transient and steadystate amplitudes will be determined for large nonlinearities and positive damping. Each chapter is written by an expert contributor in the field of nonlinear dynamics and addresses a. Oct 10, 20 for the love of physics walter lewin may 16, 2011 duration. In particular, they give an extensive reference list to find results obtained using this technique and other techniques. The duffing equation is given in the article introduction. Applications and computational simulations find, read and cite. Duffings equation is, of course, a nonlinear differential equation since it contains a term in x 3.
Download fulltext pdf on lyapunov control of duffing equation article pdf available in ieee transactions on circuits and systems i fundamental theory and applications 428. The governing differential equation of the nonlinear. The cubication and the equivalent nonlinearization methods are used to replace the original duffingharmonic oscillator by an approximate duffing equation in which the coefficients for the linear and cubic terms depend on the initial oscillation amplitude. Finally, numerical simulations using matlab are carried out to investigate the dynamic behavior such as bifurcation, chaos, and chaotic. The duffing equation or duffing oscillator, named after georg duffing 18611944, is a nonlinear secondorder differential equation used to model certain damped and driven oscillators. Here we chose the parameters so as to see chaos thanks to work of ueda in 1980. In some cases, it is also necessary to add nonlinear damping terms. In that case the behavior of the system is sensitive to the initial condition. The duffing equation with mathematica from wolfram. View enhanced pdf access article on wiley online library html view download pdf for offline viewing. Duffing equation occurs as a result of the motion of a body subjected to a nonlinear spring power, linear sticky. Equation 5 reveals that amplitude varies discontinuously with time in microphysics and tsunami motion. Pdf analytic and numerical solution for duffing equations.
Amplitudefrequency relationship to a fractional duffing. In this program helps to find the phase portraits of the duffing oscillator as well as to save the data file from which we have to plot. The analytical solutions are jacobi elliptic functions. Approximate solution for nonlinear duffing oscillator with. By using the leaf functions, the exact solutions of the cubic duffing equation can be derived under several conditions. The duffing equation wiley online books wiley online library. The duffing equation is an example of a dynamical system that exhibits chaotic behavior. Duffing s equation is, of course, a nonlinear differential equation since it contains a term in x 3. In the present paper, the duffing oscillator equation is solved using a new simple technique based on taylor theory. It is a classic example of chaos theory, where the motion of the oscillator is strongly dependent on the initial conditions. Numerical solution of duffing equation by using an improved. Duffing equation is a nonlinear differential equation, which is.
I removed the parts which are not relevant for duffing equation itself relating to quadratic stiffness terms and added the expand section template. Duffing oscillator defined by a system of differential equation 7 with no damping and with no external excitation force has one stationary point x,y 0,0 as a stable center in the case of a strong spring. The tsunami wave, in any macroscales of time, is continuous. In this paper, seven types of the exact solutions are presented based on the leaf functions. Oct 16, 2007 the forced duffing oscillator exhibits behavior ranging from limit cycles to chaos due to its nonlinear dynamics. Approximate solution for nonlinear duffing oscillator with damping. Nonlinear oscillators and their behaviour, wiley 2011, isbn.
In order to study the stability of periodic responses of the forced duffing oscillator, local stability analysis is carried out on the equations describing the slow time. In that case the route to chaos is called perioddoubling. A qualitative study of the damped dung equation and applications. The duffing equation by ivana kovacic overdrive rakuten. In this program helps to find the phase portraits of the duffing oscillator as well as to save the data file from which we have to plot in present. The general solution is a linear combination of the two eigensolutions, and for large time the.
Moreover, the duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour. Nonlinear oscillators and their behaviour brings together the results of a wealth of disseminated research literature on the duffing equation, a key engineering model with a vast number of applications in science and engineering, summarizing the findings of this research. With the numerical capabilities of mathematica, we check our solutions. Analytic and numerical solution for duffing equations. In practice one would like to understand the route to chaos in systems described by partial differential equations, such as flow in a randomly stirred fluid. A general solution of the duffing equation springerlink. Some numerical methods for obtaining harmomc duffings. The duffing equation with mathematica from wolfram library. There is no argument in adding additional nonlinear restoring force terms to describe pratical problems. This equation might represent, for example, an electric circuit or a mechanical oscillator. An alternative poincare section for steadystate responses.
Rand nonlinear vibrations 5 if det 0andtr2 4 det, then there are still two real eigenvalues, but both have the same sign as the trace tr. This python code simulates the duffing oscillator, a damped driven harmonic oscillator in a double well potential. Approximate solution of nonlinear duffing oscillator using. The duffing equation introduction we have already seen that chaotic behavior can emerge in a system as simple as the logistic map.
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