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. Pdf is not rendering correctly, you can download the pdf file here. Stochastic resonance in the duffing oscillator with matlab. The duffing oscillator is one of the prototype systems of nonlinear dynamics. We use mathematica to find the analytical solutions of the duffing problem without friction. Jan 20, 2008 boundedness and convergence of solutions of duffing s equation shiraiwa, kenichi, nagoya mathematical journal, 1977. Some numerical methods for obtaining harmomc duffings. We have suggested a numerical approach, which is based on an improved taylor matrix method, for solving duffing differential equations. 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. This quadratic equation pdf we are providing is free to download. 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 distributed as a readytorun compiled java archive. I would like to do the same thing for the frequency response, but i simply can not figure out how.
Each chapter is written by an expert contributor in the field of nonlinear dynamics and addresses a. This content was uploaded by our users and we assume good faith they have the permission to share this book. The duffing equation wiley online books wiley online library. Solutions to the oscillator equation can exhibit extreme nonlinear dynamics, including limit cycles, strange attractors, and. In the present letter, we will reveal that the problem of seeking traveling wave solutions to the hirota equation can actually be converted into solving the duffings equation for its exact solutions. A coupling successive approximation method for solving duffing equation and its application,vietnam journal of mechanics, coupling successive approximation method, solving duffing equation, coupling successive approximation method, the general duffing equation, special physical factor, free download pdf, the paper proposes an algorithm to solve a general duffing equation, in which a process of. 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. Pdf exact solution to duffing equation and the pendulum. Resonances of a harmonically forced duffing oscillator with time delay state feedback. Reference 3 4 introduces a sinusoidal timedependent power series solution. Here we chose the parameters so as to see chaos thanks to work of ueda in 1980. Approximate solution for nonlinear duffing oscillator with. Approximate periodic solutions for the helmholtzduffing equation. Nonlinear oscillators and their behaviour brings together the results of a wealth of disseminated research literature on the duffing equation, a. Bounds for the derivative of the restoring force are given which ensure the existence and uniqueness of a. A general solution of the duffing equation springerlink.
The influence of noise on the duffing oscillators frequency response is. This file contains additional information such as exif metadata which may have been added by the digital camera, scanner, or software program used to create or digitize it. Analytic and numerical solution for duffing equations. Analytical and numerical investigation on the duffing. The duffing equation with mathematica from wolfram library. Basins of attraction for the duffing equation with no forcing term. Also, the conclusion of bifurcation of the parameterdependent system has been drawn numerically. Moser proposed independently the boundedness problem for solutions of duffing s equation. A qualitative study of the damped dung equation and. A discretization method is applied in this paper to the forced duffing oscillator with fractionalorder damping. Discretization of forced duffing system with fractionalorder.
Moreover, the duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour. Also, a numerical solution is realized and compared with the analytical one. Kawakamiqualitative study on the solutions of duffings equation, thesis 1973, kyoto university. I dont see how you are passing the values of the parameters e. It is an example of a dynamical system that exhibits chaotic behavior. Numerical solution of duffing equation by using an improved. In that case the route to chaos is called perioddoubling. The duffing oscillator moves in a double well potential, sometimes characterized as nonlinear elasticity, with sinusoidal external forcing.
A coupling successive approximation method for solving. The method is based on the approximation by the truncated taylor series about center zero. Pdf analytic and numerical solution for duffing equations. Duffing equation, has a standard wellknown exact solution 1. Pdf daftardar gejji and hossein jafari have proposed a new iterative. Oct 16, 2007 the forced duffing oscillator exhibits behavior ranging from limit cycles to chaos due to its nonlinear dynamics. If the file has been modified from its original state, some details such as the timestamp may not fully reflect those of the original file. The graph illustrates what we expected about the period. It is a classic example of chaos theory, where the motion of the oscillator is strongly dependent on the initial conditions. Thywle 1 to analyze the forced oscillation of a helmholtzduffing oscillator. Most important quadratic equation question pdf with answers.
Numerical solution of the duffing equation with random. It computes the averaged signal and noise amplitude spectra for varying noise strength. Abstract duffing oscillator or duffing equation is one of the most significant and classical nonlinear ordinary differential equations in view of its diverse. Some numerical methods for obtaining harmonic and subharmonic solutions of duffings equation by m. With the numerical capabilities of mathematica, we check our solutions. Duffing equation it resolves in many of new process in. Exact solution to duffing equation and the pendulum.
I am not very mathematically advanced so the first. On duffings equation journal of applied mechanics asme. Amplitudefrequency relationship to a fractional duffing oscillator. Sinusoidal timedependent power series solution to modified. The duffing equation is an example of a dynamical system that exhibits chaotic behavior. Analytical solution of strongly nonlinear duffing oscillators. How to plot the basin of attraction of a forced duffing.
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 mechanical response in deterministic situation is described by the duffing equation, whose numerical solution is obtained with the rungekuttafehlenberg algorithm, while probabilistic analysis is carried out using the generalized stochastic perturbation technique enriched with automatic optimization of the approximating polynomial. Ejs duffing oscillator model was created using the easy java simulations ejs modeling tool. Lecture notes on nonlinear vibrations cornell university. The approximate solving methods for the cubic duffing equation based. Resonances of a harmonically forced duffing oscillator with time. The duffing map also called as holmes map is a discretetime dynamical system. Applying this method successfully we investigate the approximate solution of the modified duffing equations, for n 4 and 5.
The duffing equation introduction we have already seen that chaotic behavior can emerge in a system as simple as the logistic map. Analytical solutions of undamped and autonomous cubicquintic. In the present letter, we will reveal that the problem of seeking traveling wave solutions to the hirota equation can actually be converted into solving the duffing s equation for its exact solutions. Boundedness and convergence of solutions of duffings equation shiraiwa, kenichi, nagoya mathematical journal, 1977. Pdf numerical solution of duffing equation by using an improved. 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. Moser proposed independently the boundedness problem for solutions of duffings equation. From this solution, the maximum transient and steadystate amplitudes will be determined for large nonlinearities and positive damping. Approximate solutions to this equation also are available 2. Oct 14, 20 numerically solve the driven, damped, duffing oscillator with noise. For this, the duffing equation is solved numerically and then, the spectrograms of vibrations are obtained through a discretetime fourier transform. 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 analytical solutions are jacobi elliptic functions.
The duffing map takes a point x n, y n in the plane and maps it to a new point given by. Harmonic and subharmonic solutions are determined numerically for the forced oscillations of a. Forced duffing oscillator file exchange matlab central. Existence and global exponential stability of almost periodic solutions for a class of delay duffing equations on time scales yang, li and li, yongkun, abstract and applied analysis, 2014. Im having a hard time to find a scenario in physics or in any science where i can meet the duffing equation with all coefficients not zero the most general case. In this work we obtain an exact solution to cubic duffing oscillator equation. Approximate solution for nonlinear duffing oscillator with damping. The dynamics of the discretized fractionally damped duffing equation has been examined numerically.
When the duffing oscillator involves damping, the amplitude of oscillation reduces. This python code simulates the duffing oscillator, a damped driven harmonic oscillator in a double well potential. A qualitative study of the damped dung equation and applications. Experiments with duffing oscillator ucsd mathematics. For this, the duffing equation is solved numerically and then, the spectrograms of vibrations are. How to plot the basin of attraction of a forced duffing equation using mathematica. The forced duffing oscillator exhibits behavior ranging from limit cycles to chaos due to its nonlinear dynamics. Softening duffing oscillator reliability assessment subject to evolutionary stochastic excitation asceasme journal of risk and uncertainty in engineering systems, part a. The nonlinear helmholtz equation has received lots of attention especially in the last decade. Mathematica could be used to bring students to use jacobi elliptic functions without effort. We use this solution to obtain the solution to pendulum equation. Analytical solution derived clarifies how the amplitude modulation occurs.
The approximate solving methods for the cubic duffing equation. Effects of noise on the frequency response of the monostable. Stability analysis of neutral stochastic delay differential equations by a generalisation of banachs contraction principle. Numerical solution of duffing equation by using an improved taylor matrix method. Pdf analytical solution of strongly nonlinear duffing. Experiments with the duffing oscillator from nonlinear dynamics and chaos by j. Numerically solve the driven, damped, duffing oscillator with noise. In that case the behavior of the system is sensitive to the initial condition. We consider a secondorder equation of duffing type. Exact solution to du ng equation and the pendulum equation. Finally, a new approximate solution of the intermediateintegral of the damped duffing oscillator with weak damping isconstructed. Moreover, the solutions of the nonlinear duffing oscillator with the damping effect are. How to graph the duffing frequency response physics forums.
258 321 986 127 1305 1201 96 570 534 43 360 1232 353 478 518 911 629 1034 949 1195 436 931 261 513 531 949 661 485 1105 120 924 797 763 1474 1401 1147 477 692 1130 545 469 567 484 1230 1099 794