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