Carolina Pascual

Exact solution for the nonlinear pendulum

This paper deals with the nonlinear oscillation of a simple pendulum and presents not only the exact formula for the period but also the exact expression of the angular displacement as a function of the time, the amplitude of oscillations and the angular frequency for small oscillations. This angular displacement is written in terms of the Jacobi elliptic function sn(u;m) using the following initial conditions: the initial angular displacement is different from zero while the initial angular velocity is zero. The angular displacements are plotted using Mathematica, an available symbolic computer program that allows us to plot easily the function obtained. As we will see, even for amplitudes as high as 0.75p (135o) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency w in terms of the complete elliptic integral of the first kind. We can conclude that for amplitudes lower than 135o the periodic motion exhibited by a simple pendulum is practically harmonic but its oscillations are not isochronous (the period is a function of the initial amplitude). We believe that present study may be a suitable and fruitful exercise for teaching and better understanding the behavior of the nonlinear pendulum in advanced undergraduate courses on classical mechanics.

Higher-order analytical approximate solutions to the nonlinear pendulum by He's homotopy method

A modified Hes homotopy perturbation method is used to calculate the periodic solutions of a nonlinear pendulum. The method has been modified by truncating the infinite series corresponding to the first-order approximate solution and substituting a finite number of terms in the second-order linear differential equation. As can be seen, the modified homotopy perturbation method works very well for high values of the initial amplitude. Excellent agreement of the analytical approximate period with the exact period has been demonstrated not only for small but also for large amplitudes A (the relative error is less than 1% for A < 152°). Comparison of the result obtained using this method with the exact ones reveals that this modified method is very effective and convenient.

Application of He's Homotopy Perturbation Method to the Relativistic (An)harmonic Oscillator. II: A More Accurate Approximate Solution

The homotopy perturbation method is used to solve the nonlinear differential equation that governs the behaviour of a relativistic oscillator for which the nonlinearity (anharmonicity) is a relativistic effect due to the time line dilation along the world line. The approximate formulas obtained show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation A. Only one iteration leads to high accuracy of the solutions and for any value of A and the discrepancy between the approximate frequency and the exact one never exceeds 1.6%.

Higher-order approximate solutions to the relativistic and Duffing-harmonic oscillators by modified He's homotopy methods

A modified Hes homotopy perturbation method is used to calculate higher-order analytical approximate solutions to the relativistic and Duffing-harmonic oscillators. The Hes homotopy perturbation method is modified by truncating the infinite series corresponding to the first-order approximate solution before introducing this solution in the second-order linear differential equation, and so on. We find this modified homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. The approximate formulae obtained show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation, including the limiting cases of amplitude approaching zero and infinity. For the relativistic oscillator, only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 1.6% for small and large values of oscillation amplitude, while this relative error is 0.65% for two iterations with two harmonics and as low as 0.18% when three harmonics are considered in the second approximation. For the Duffing-harmonic oscillator the relative error is as low as 0.078% when the second approximation is considered. Comparison of the result obtained using this method with those obtained by the harmonic balance methods reveals that the former is very effective and convenient.

Mixed phase-amplitude holographic gratings recorded in bleached silver halide materials

The coupled wave theory of Kogelnik has given a well-established basis for the comprehension of how light propagates inside a volume hologram. This theory gives a good approximation for the diffraction efficiency of both volume phase holograms and volume absorption holograms. Mixed holograms (phase and absorption) have also been dealt with from the point of view of the coupled wave theory. In this paper we use Kogelniks coupled wave theory to give quantitative information about the mechanisms which produce mixed gratings in photographic emulsions. In particular, we demonstrate that mixed amplitude-phase gratings are recorded on photographic emulsions when fixation-free bleaching techniques are used to obtain volume holograms. We will prove that the oxidation products of the bleach can give rise to an absorption modulation at high values of exposure and high concentrations of potassium bromide in the bleach bath. We will also give quantitative data regarding the absorption created by these oxidation products.

Approximate analytical solutions for the relativistic oscillator using a linearized harmonic balance method

The analytical approximate technique developed by Wu et al. for conservative oscillators with odd nonlinearity is used to construct approximate frequency-amplitude relations and periodic solutions to the relativistic oscillator. By combining Newtons method with the method of harmonic balance, analytical approximations to the oscillation period and periodic solutions are constructed for this oscillator. The approximate periods obtained are valid for the complete range of oscillation amplitudes, A, and the discrepancy between the second approximate period and the exact one never exceeds 1.24%, and it tends to 1.09% when A tends to infinity. Excellent agreement of the approximate periods and periodic solutions with the exact ones are demonstrated and discussed.

Closed-Form Exact Solutions for the Unforced Quintic Nonlinear Oscillator

Closed-form exact solutions for the periodic motion of the one-dimensional, undamped, quintic oscillator are derived from the first integral of the nonlinear differential equation which governs the behaviour of this oscillator. Two parameters characterize this oscillator: one is the coefficient of the linear term and the other is the coefficient of the quintic term. Not only the common case in which both coefficients are positive but also all possible combinations of positive and negative values of these coefficients which provide periodic motions are considered. The set of possible combinations of signs of these coefficients provides four different cases but only three different pairs of period-solution. The periods are given in terms of the complete elliptic integral of the first kind and the solutions involve Jacobi elliptic function. Some particular cases obtained varying the parameters that characterize this oscillator are presented and discussed. The behaviour of the periods as a function of the initial amplitude is analysed and the exact solutions for several values of the parameters involved are plotted. An interesting feature is that oscillatory motions around the equilibrium point that is not at are also considered.

Exact and approximate solutions for the anti-symmetric quadratic truly nonlinear oscillator

The exact solution of the anti-symmetric quadratic truly nonlinear oscillator was expressed as a piecewise function.The Fourier coefficients of the exact solution were computed numerically and we showed these decrease rapidly.Using just a few of Fourier coefficients provides an accurate analytical representation of the exact periodic solution.Analytical approximate solutions are built up containing only two harmonics as well as a rational harmonic representation.The two-harmonic representation is more accurate than the rational harmonic representation. The exact solution of the anti-symmetric quadratic truly nonlinear oscillator is derived from the first integral of the nonlinear differential equation which governs the behavior of this oscillator. This exact solution is expressed as a piecewise function including Jacobi elliptic cosine functions. The Fourier series expansion of the exact solution is also analyzed and its coefficients are computed numerically. We also show that these Fourier coefficients decrease rapidly and, consequently, using just a few of them provides an accurate analytical representation of the exact periodic solution. Some approximate solutions containing only two harmonics as well as a rational harmonic representation are obtained and compared with the exact solution.

Finite difference time domain method (FDTD) to predict the efficiencies of the different orders inside a volume grating

Different electromagnetic theories have been applied in order to understand the interaction of the electromagnetic radiation with diffraction gratings. Kogelniks Coupled Wave Theory, for instance, has been applied with success to describe the diffraction properties of sinusoidal volume gratings. Nonetheless the predictions of Kogelniks theory deviate from the actual behaviour whenever the hologram is thin or the refractive index is high. In these cases, it is necessary to use a more general Coupled Wave Theory (CW) or the Rigorous Coupled Wave Theory (RCW). Both of these theories allow for more than two orders propagating inside the hologram. On the other hand, there are some methods that have been used long in different physical situations, but with relatively low application in the field of holography. This is the case of the finite difference in the temporal domain (FDTD) method to solve Maxwell equations. In this work we present an implementation of this method applied to volume holographic diffraction gratings.

Comparison of electromagnetic theories to predict the efficiencies of the different orders inside a volume grating

In this work we make a comparative study between different theories to predict the efficiencies of the different orders that propagate inside a volume phase grating. For the case of a pure sinusoidal grating, transmission and reflection, the theories of Rigorous Coupled Wave Theory (RCW), Coupled Wave Theory (CW) and Kogelniks Theory are compared. This comparison allows establishing the range of physical values where the more approximate theories, CW and Kogelnik´s Theory are applicable. On the other hand for the case of a general dielectric grating, transmission and reflection, the RCW, CW, and a thin matrix decomposition method (TMDM) are also compared. The theoretical study is also validated by comparing the theoretical results with experimental data obtained in volume phase diffraction gratings recorded on photographic emulsions. To record the volume phase diffraction gratings BB-640 emulsions were exposed to an interference pattern of light from a He-Ne laser (633 nm).