Numerical methods in classical mechanics: differential equations

Journal of Sound and Vibration, 2005

In the given write up I shall be giving a theoretical and historical overview of classical mechanics. After that I will be discussing about one family of one dimensional maps with their universality, similarity and apparantly randomness in order to get a glimpse of chaos in one dimensional systems. .

Computer Methods in Applied Mechanics and Engineering, 2007

A novel form of an explicit numeric-analytic technique is developed for solving strongly nonlinear oscillators of engineering interest. The analytic part of this technique makes use of Adomian Decomposition Method (ADM), but unlike other analytical solutions it does not rely on the functional form of the solution over the whole domain of the independent variable. Instead it discretizes the domain and solves multiple IVPs recursively. ADM uses a rearranged Taylor series expansion about a function and finds a series of functions which add up to generate the required solution. The present method discretizes the axis of the independent variable and only collects lower powers of the chosen step size in series solution. Each function constituting the series solution is found analytically. It is next shown that the modified ADM can be used to obtain the analytical solution,in a piecewise form. For nonlinear oscillators such a piecewise solution is valid only within a chosen time step. An attempt has been made to address few issues like the order of local error and convergence of the method. Emphasis has been on the application of the present method to a number of well known oscillators. The method has the advantage of giving a functional form of the solution within each time interval thus one has access to finer details of the solution over the interval. This is not possible in purely numerical techniques like the Runge-Kutta method, which provides solution only at the two ends of a given time interval, provided that the interval is chosen small enough for convergence. It is shown that the present technique successfully overcomes many limitations of the conventional form of ADM. The present method has the versatility and advantages of numerical methods for being applied directly to highly nonlinear problems and also have the elegance and other benefits of analytical techniques.

International Journal of Bifurcation and Chaos, 2016

The chaotic behavior of the modified Rayleigh–Duffing oscillator with [Formula: see text] potential and external excitation is investigated both analytically and numerically. The so-called oscillator models, for example, ship rolling motions. The single well and triple well potential cases are considered. Melnikov method is applied and the conditions for the existence of homoclinic and heteroclinic chaos are obtained. The effects of nonlinear damping on roll motion of ships are analyzed in detail. As it is known, nonlinear roll damping is a very important parameter in estimating ship responses. It is noted that the pure and unpure quadratic damping parameters affect the Melnikov criterion in the heteroclinic and homoclinic cases respectively while the pure cubic parameter affects the amplitude in both cases. The predictions have been tested with numerical simulations based on the basin of attraction. It is pointed out that certain quadratic damping effects are contrary to cubic damp...

2014

Solution of non-linear dynamic systems is dependent on exact knowledge of the initial conditions. Even a slight deviation of these values can cause substantial change in the overall course of the event. It then appears chaotic. Development of such dynamic system can be represented using abstract phase space through attractors, fractals, etc. A typical example is Lorenz attractor, which is in a three-dimensional view shaped as two intertwined spirals. Rössler attractor is a relatively simple system on which chaos in geometric form can be shown in a time sequence. Among the non-traditional oscillators in non-linear mechanics can be classified Duffing and Van der Pol oscillators. This paper shows an example of a chaotic attractor formed in a non-periodic mode, obtained in an experiment of water dripping from an unclosed valve.

Journal of Sound and Vibration, 1996

American Journal of Mechanical Engineering, 2014

Solution of non-linear dynamic systems is dependent on exact knowledge of the initial conditions. Even a slight deviation of these values can cause substantial change in the overall course of the event. It then appears chaotic. Development of such dynamic system can be represented using abstract phase space through attractors, fractals, etc. A typical example is Lorenz attractor, which is in a three-dimensional view shaped as two intertwined spirals. Rössler attractor is a relatively simple system on which chaos in geometric form can be shown in a time sequence. Among the non-traditional oscillators in non-linear mechanics can be classified Duffing and Van der Pol oscillators. This paper shows an example of a chaotic attractor formed in a non-periodic mode, obtained in an experiment of water dripping from an unclosed valve.

1999

This thesis is concerned with the study, classically and quantum mechanically, of the square billiard with particular attention to chaos in both cases. Classically, we show that the rotating square billiard has two regular limits with a mixture of order and chaos between, depending on an energy parameter, E. This parameter ranges from -2a;^ to oo, where u is the angular rotation, corresponding to the two integrable limits. The rotating square billiard has simple enough geometry to permit us to elucidate that the mechanism for chaos with rotation or curved trajectories is not flyaway, as previously suggested, but rather the accumulation of angular dispersion from a rotating line. Furthermore, we find periodic cycles which have asymmetric trajectories, below the value of E at which phase space becomes disjointed. These trajectories exhibit both left and right hand curvatures due to the fine balance between Centrifugal and Coriolis forces. Quantum mechanically, we compare the spectral analysis results for the square billiard with three different theoretical distribution functions. A new feature in the study is the correspondence we find, by utilising the Berry-Robnik parameter q, between classical E and a quantum rotation parameter cu. The parameter q gives the ratio of chaotic quantum phase volume which we can link to the ratio of chaotic phase volume found classically for varying values of E. We find good correspondence, in particular, the different values of ^ as a; is varied refiect the births and subsequent destructions of the different periodic cycles. We also study wave packet dynamics, necessitating the adaptation of a one dimensional unitary integration method to the two dimensional square billiard. In concluding we suggest how this work may be used, with the aid of the chaotic phase volumes calculated, in future directions for research work. n Declaration

We illustrate a procedure based on the Magnus expansion for studying mechanical problems which lead to non-autonomous systems of linear ODE's. The effectiveness of the Magnus method is enlighten by the analysis of a bifurcation problem in the framework of three-dimensional non-linear elasticity. In particular, for an isotropic compressible elastic tube subject to an azimuthal shear primary deformation we study the possibility of axially periodic twist-like bifurcations. The approximate matricant of the resulting differential problem and the first singular value of the bifurcating load corresponding to a non-trivial bifurcation are determined by employing a simplified version of the Magnus method, characterized by a truncation of the Magnus series after the second term.

Chaos Theory - Modeling, Simulation and Applications - Selected Papers from the 3rd Chaotic Modeling and Simulation International Conference (CHAOS2010), 2011

We study Melnikov conditions predicting appearance of chaos in Duffing oscillator with hardening type of non-linearity under two-frequency excitation acting in the vicinity of the principal resonance. Since Hamiltonian part of the system contains no saddle points, Melnikov method cannot be applied directly. After separating the external force into two parts, we use a perturbation analysis that allows recasting the original system to the form suitable for Melnikov analysis. At the initial step, we perform averaging at one of the frequencies of the external force. The averaged equations are then analyzed by traditional Melnikov approach, considering the second frequency component of the external force and the dissipation term as perturbations. The numerical study of the conditions for homoclinic bifurcation found by Melnikov theory is performed by varying the control parameters of amplitudes and frequencies of the harmonic components of the external force. The predictions from Melnikov theory have been further verified numerically by integrating the governing differential equations and finding areas of chaotic behavior. Mismatch between the results of theoretical analysis and numerical experiment is discussed.