Approximate solution for nonlinear duffing oscillator with. The duffing equation wiley online books wiley online library. In that case the behavior of the system is sensitive to the initial condition. Exact solutions to cubic duffing equation by leaf functions. Approximate solution for the duffingharmonic oscillator. Forced duffing oscillator file exchange matlab central. Duffing s equation is, of course, a nonlinear differential equation since it contains a term in x 3. This python code simulates the duffing oscillator, a damped driven harmonic oscillator in a double well potential. Analytic and numerical solution for duffing equations.
Finally, numerical simulations using matlab are carried out to investigate the dynamic behavior such as bifurcation, chaos, and chaotic. Mathematica could be used to bring students to use jacobi elliptic functions without effort. It describes the forced oscillations of several mechanical and electrical systems e. The duffing equation with mathematica from wolfram library. 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. Solutions to the oscillator equation can exhibit extreme nonlinear dynamics, including limit cycles, strange attractors, and. Numerical solution of duffing equation by using an. In that case the route to chaos is called perioddoubling. The duffing equation by ivana kovacic overdrive rakuten. Applications and computational simulations find, read and cite. The duffing oscillator moves in a double well potential, sometimes characterized as nonlinear elasticity, with sinusoidal external forcing. A general solution of the duffing equation springerlink.
The duffing equation is an example of a dynamical system that exhibits chaotic behavior. 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. Amplitudefrequency relationship to a fractional duffing oscillator. The duffing equation is given in the article introduction. Each chapter is written by an expert contributor in.
However, there are simply to many extensions to mention them all. The general solution is a linear combination of the two eigensolutions, and for large time the. This is a matlab gui project,very simple for draw a piecewise function. The duffing oscillator is one of the prototype systems of nonlinear dynamics. This content was uploaded by our users and we assume good faith they have the permission to share this book. Duffing equation which called japanese attractor or ueda attractor. The duffing oscillator is a common model for nonlinear phenomena in science and engineering. Moreover, the solutions of the nonlinear duffing oscillator with the damping effect are obtained using the fourthorder. The cubication and the equivalent nonlinearization methods are used to replace the original duffing harmonic oscillator by an approximate duffing equation in which the coefficients for the linear and cubic terms depend on the initial oscillation amplitude. Amplitudefrequency relationship to a fractional duffing.
The forced duffing oscillator exhibits behavior ranging from limit cycles to chaos due to its nonlinear dynamics. 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. A qualitative study of the damped dung equation and applications. 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. Analogue electrical circuit for simulation of the duffing. Experiments with duffing oscillator ucsd mathematics. Experiments with the duffing oscillator from nonlinear dynamics and chaos by j. An alternative poincare section for steadystate responses. Duffing equation is a nonlinear differential equation, which is. It is a classic example of chaos theory, where the motion of the oscillator is strongly dependent on the initial conditions.
In some cases, it is also necessary to add nonlinear damping terms. When the periodic force that drives the system is large, chaotic behavior emerges and the phase space diagram is a strange attractor. The tsunami wave, in any macroscales of time, is continuous. It is an example of a dynamical system that exhibits chaotic behavior. In addition, secondary resonance corresponding to strong hard excitation is also discussed. 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. Approximate solution of nonlinear duffing oscillator using. There is no argument in adding additional nonlinear restoring force terms to describe pratical problems.
I dont see how you are passing the values of the parameters e. Each chapter is written by an expert contributor in the field of nonlinear dynamics and addresses a. 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. The duffing equation with mathematica from wolfram. Attractors in duffing equation mathematics stack exchange. Pdf on jan 10, 2017, j sunday and others published the duffing oscillator. 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. Fixed points and their stability are discussed for the discrete system obtained. In this paper, seven types of the exact solutions are presented based on the leaf functions. Moreover, the duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour.
Oct 10, 20 for the love of physics walter lewin may 16, 2011 duration. In i kovacic and m j brennan eds, the duffing equation. Duffings equation an overview sciencedirect topics. From this solution, the maximum transient and steadystate amplitudes will be determined for large nonlinearities and positive damping. 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. 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. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients. 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. Equation 5 reveals that amplitude varies discontinuously with time in microphysics and tsunami motion. This equation might represent, for example, an electric circuit or a mechanical oscillator. To simplify the computations that follow, use parametricndsolvevalue with only the initial conditions as parameters and other values fixed. The governing differential equation of the nonlinear. It can travel, for example, from some a place in the pacific to some a coast.
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. Here we chose the parameters so as to see chaos thanks to work of ueda in 1980. Discretization of forced duffing system with fractionalorder. The oscillator can be used to model rotor subjected to nonlinear damping and stiffness with external excitation. I removed the parts which are not relevant for duffing equation itself relating to quadratic stiffness terms and added the expand section template. Stationary baseband solutions of the equation come about as a balance between nonlinearity and dispersion, and thus represent solitary wave solutions to the system.
University of novi sad, faculty of technical sciences, serbia. Forced harmonic vibration of a duffing oscillator with linear. Lecture notes on nonlinear vibrations cornell university. Amplitudefrequency relationship to a fractional duffing oscillator arising in microphysics and. 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. Numerical solution of duffing equation by using an improved. In particular, they give an extensive reference list to find results obtained using this technique and other techniques. 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. Nonlinear oscillators and their behaviour, wiley 2011, isbn. Some numerical methods for obtaining harmomc duffings. View enhanced pdf access article on wiley online library html view download pdf for offline viewing. Approximate solution for nonlinear duffing oscillator with damping. 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.
Approximate solution for the duffingharmonic oscillator by. The method is based on the approximation by the truncated taylor series about center zero. We use mathematica to find the analytical solutions of the duffing problem without friction. Duffings equation is, of course, a nonlinear differential equation since it contains a term in x 3. We have suggested a numerical approach, which is based on an improved taylor matrix method, for solving duffing differential equations. In the present paper, the duffing oscillator equation is solved using a new simple technique based on taylor theory. The analytical solutions are jacobi elliptic functions. Duffing equation occurs as a result of the motion of a body subjected to a nonlinear spring power, linear sticky. The duffing map also called as holmes map is a discretetime dynamical system. New in mathematica 9 parametric differential equations sensitivity of the duffing equation explore the sensitivity of the duffing equation to initial conditions.
The duffing equation introduction we have already seen that chaotic behavior can emerge in a system as simple as the logistic map. I think the actual values of those parameters which you dont supply may also give some idea about why ode45 thinks the solution is linear over. Pdf analytic and numerical solution for duffing equations. 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 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. Oct 16, 2007 the forced duffing oscillator exhibits behavior ranging from limit cycles to chaos due to its nonlinear dynamics. If tr 0, then both eigenvalues are positive and the solution becomes unbounded as t goes to in. Pdf exact solution to duffing equation and the pendulum. Some numerical methods for obtaining harmomc duffings equation. The duffing map takes a point x n, y n in the plane and maps it to a new point given by. 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. With the numerical capabilities of mathematica, we check our solutions. By using the leaf functions, the exact solutions of the cubic duffing equation can be derived under several conditions.
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