A. Hernández

APPLICATION OF THE HOMOTOPY PERTURBATION METHOD TO THE NONLINEAR PENDULUM

The homotopy perturbation method is used to solve the nonlinear differential equation that governs the nonlinear oscillations of a simple pendulum, and an approximate expression for its period is obtained. Only one iteration leads to high accuracy of the solutions and the relative error for the approximate period is less than 2% for amplitudes as high as 130°. Another important point is that this method provides an analytical expression for the angular displacement as a function of time as the sum of an infinite number of harmonics; although for practical purposes it is sufficient to consider only a finite number of harmonics. We believe that the present study may be a suitable and fruitful exercise for teaching and better understanding perturbation techniques in advanced undergraduate courses on classical mechanics.

Hand gesture recognition following the dynamics of a topology-preserving network

We present a new structure capable of characterizing hand posture, as well as its movement. Topology of a self-organizing neural network determines posture, whereas its adaptation dynamics throughout time determines gesture. This adaptive character of the network allows us to avoid the correspondence problem of other methods, so that the gestures are modelled by the movement of the neurons. To validate this method, we have trained the system with 12 gestures, some of which are very similar, and have obtained high success rates (over 97%). This application of a self-organizing network opens up a new field of research because its topology is used to characterize the objects and not to classify them, as is usually the case.

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.

An improved 'heuristic' approximation for the period of a nonlinear pendulum: linear analysis of a classical nonlinear problem

This work was supported by the “Ministerio de Educacion y Ciencia”, Spain, under project FIS2005-05881-C02-02, and by the “Generalitat Valenciana”, Spain, under project ACOMP/2007/020.

Approximation for a large-angle simple pendulum period

An approximation scheme to obtain the period for large amplitude oscillations of a simple pendulum is analysed and discussed. The analytical approximate formula for the period is the same as that suggested by Hite (2005 Phys. Teach. 43 290), but it is now obtained analytically by means of a term-by-term comparison of the power-series expansion for the approximate period with the corresponding series for the exact period.

Analysis of metals and phosphorus in biodiesel B100 from different feedstock using a Flow Blurring® multinebulizer in inductively coupled plasma-optical emission spectrometry.

A simple and fast method for determining the content of Na, K, Ca, Mg, P, and 20 heavy metals in biodiesel samples with inductively coupled plasma optical emission spectrometry (ICP OES) using a two-nozzle Flow Blurring(®) multinebulizer prototype and on-line internal standard calibration, are proposed. The biodiesel samples were produced from different feedstock such as sunflower, corn, soybean and grape seed oils, via a base catalyst transesterification. The analysis was carried out without any sample pretreatment. The standards and samples were introduced through one of the multinebulizer nozzles, while the aqueous solution containing yttrium as an internal standard was introduced through the second nozzle. Thus, the spectral interferences were compensated and the formation of carbon deposits on the ICP torch was prevented. The determination coefficients (R(2)) were greater than 0.99 for the studied analytes, in the range 0.21-14.75 mg kg(-1). Short-term and long-term precisions were estimated as relative standard deviation. These were acceptable, their values being lower than 10%. The LOQ for major components such as Ca, K, Mg, Na, and P, were within a range between 4.9 ng g(-1) for Mg (279.553 nm) and 531.1 ng g(-1) for Na (588.995 nm), and for the other 20 minor components they were within a range between 1.1 ng g(-1) for Ba (455.403 nm) and 2913.9 ng g(-1) for Pb (220.353 nm). Recovery values ranged between 95% and 106%.

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.

Characterization and synthesis of objects using growing neural gas

In this article it is made a study of the characterization capacity and synthesis of objects of the self-organizing neural models. These networks, by means of their competitive learning, try to preserve the topology of an input space. This capacity is being used for the representation of objects and their movement with topology preserving networks. We characterized the object to represent by means of the obtained maps and kept information solely on the coordinates and the colour from the neurons. From this information it is made the synthesis of the original images, applying mathematical morphology and simple filters on the information which it is had.

Thin and thick diffraction gratings: Thin matrix decomposition method

Summary A brief review of the properties of transmission diffraction gratings is presented. Two types of gratings will be analyzed: thin and volume gratings explaining how the efficiency of the different orders that propagate inside the gratings can be calculated in both cases. For thin diffraction gratings the so-called amplitude transmittance method is applied in order to get the amplitude of the different orders, whereas in the case of volume gratings more complex methods are needed, such as Coupled Wave or modal theories. We will comment on the thin matrix decomposition method (TMDM), firstly proposed by Alferness, which gives a very intuitive approach and connects the properties of thin gratings to the properties of volume ones. The thin matrix decomposition method consists in dividing the volume grating in a number of thin gratings and applying the amplitude transmittance method to each thin grating. In this way the output of a grating will be considered as the input of the next and any individual grating can be treated by the amplitude transmittance method. The novelty of this work is that a comparison is made between the analytical expressions obtained by Alferness using the TMDM with the numerical results obtained using the coupled wave (CW) and rigorous coupled wave (RCW) theories for the efficiencies of the zero, first and second order when a plane wave incides onto a sinusoidal diffracion grating at the second on-Bragg replay angular condition.

Reply to 'Comment on "Approximation for the large-angle simple pendulum period"'

In their comment, Yuan and Ding derived another analytical approximate expression for the large-angle pendulum period, which they compare with other expressions previously published. Most of these approximate formulas are based on the approximation of the original nonlinear differential equation for the simple pendulum motion. However, we point out that another procedure is possible to obtain an approximate expression for the period. This procedure is based on the approximation of the exact period formula—which is expressed in terms of a complete elliptic integral of the first kind—instead of the approximation of the original differential equation. This last procedure is used, for example, by Carvalhaes and Suppes using the arithmetic–geometric mean.