Tarsicio Beléndez

Large and small deflections of a cantilever beam

The classical problem of the deflection of a cantilever beam of linear elastic material, under the action of an external vertical concentrated load at the free end, is analysed. We present the differential equation governing the behaviour of this physical system and show that this equation, although straightforward in appearance, is in fact rather difficult to solve due to the presence of a non-linear term. In this sense, this system is similar to another well known physical system: the simple pendulum. An approximation of the behaviour of a cantilever beam for small deflections was obtained from the equation for large deflections, and we present various numerical results for both cases. Finally, we compare the theoretical results with the experimental results obtained in the laboratory.

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.

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.

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.

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.

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.

Determination of the refractive index and thickness of holographic silver halide materials by use of polarized reflectances

A method to determine the refractive index and thickness of silver halide emulsions used in holography is presented. The emulsions are in the form of a layer of film deposited on a thick glass plate. The experimental reflectances of p-polarized light are measured as a function of the incident angles, and the values of refractive index, thickness, and extinction coefficient of the emulsion are obtained by using the theoretical equation for reflectance. As examples, five commercial holographic silver halide emulsions are analyzed. The procedure to obtain the measurements and the numerical analysis of the experimental data are simple, and agreement of the calculated reflectances, by use of the thickness and refractive index obtained, with the measured reflectances is satisfactory.

An analysis of the classical Doppler effect

The Doppler effect is a phenomenon which relates the frequency of the harmonic waves generated by a moving source with the frequency measured by an observer moving with a different velocity from that of the source. The classical Doppler effect has usually been taught by using a diagram of moving spheres (surfaces with constant phase) centred at the source. This method permits an easy and graphical interpretation of the physics involved for the case in which the source moves with a constant velocity and the observer is at rest, or the reciprocal problem (the source is at rest and the observer moves). Nevertheless it is more difficult to demonstrate, by this method, the relation of the frequencies for a moving source and observer. We present an easy treatment where the Doppler formulae are obtained in a simple way. Different particular cases will be discussed by using this treatment.

Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions

Accurate approximate closed-form solutions for the cubic-quintic Duffing oscillator are obtained in terms of elementary functions. To do this, we use the previous results obtained using a cubication method in which the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a cubic Duffing equation. Explicit approximate solutions are then expressed as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function cn. Then we obtain other approximate expressions for these solutions, which are expressed in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean is used and the rational harmonic balance method is applied to obtain the periodic solution of the original nonlinear oscillator.

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.