Augusto 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.

Optimization of a thick polyvinyl alcohol-acrylamide photopolymer for data storage using a combination of angular and peristrophic holographic multiplexing

The capability of polyvinyl alcohol-acrylamide photopolymer materials to obtain angularly multiplexed holographic gratings has been demonstrated [Appl. Phys. B 76, 851 (2003)]. A combination of two multiplexing methods--peristrophic and angular multiplexing--is used to record 60 holograms. An exposure schedule method is used to optimize the capability of the photopolymerizable holographic material and obtain holograms with a higher, more uniform diffraction efficiency. In addition, because of this exposure schedule method, the entire dynamic range (M#) of the material will be exploited, obtaining values of approximately M# approximately 9 in layers approximately 800 microm thick.

Physical and effective optical thickness of holographic diffraction gratings recorded in photopolymers

In recent years the interest in thick holographic recording materials for storage applications has increased. In particular, photopolymers are interesting materials for obtaining inexpensive thick dry layers with low noise and high diffraction efficiencies. Nonetheless, as will be demonstrated in this work, the attenuation in depth of light during the recording limits dramatically the effective optical thickness of the material. This effect must be taken into account whenever thick diffraction gratings are recorded in photopolymer materials. In this work the differences between optical and physical thickness are analyzed, applying a method based on the Rigorous Coupled Wave Theory and taking into account the attenuation in depth of the refractive index profile. By doing this the maximum optical thickness that can be achieved can be calculated. When the effective thickness is known, then the real storage capacity of the material can be obtained.

New photopolymer holographic recording material with sustainable design.

Photopolymers that absorb in the visible spectrum are useful for different applications such as in the development of holographic memories, holographic optical elements or as holographic recording media. Photopolymers have an undesirable feature, the toxicity of their components and their low environmental compatibility, particularly if we analyse the life cycle of the devices made with these materials and their interaction with the environment. In this work we developed a new photopolymer with photochemical and holographic features similar to those of the standard material but with an improved design from the environmental point of view.

3 Dimensional analysis of holographic photopolymers based memories

One of the most interesting applications of photopolymers is as holographic recording materials for holographic memories. One of the basic requirements for this application is that the recording material thickness must be 500 microm or thicker. In recent years many 2-dimensional models have been proposed for the analysis of photopolymers. Good agreement between theoretical simulations and experimental results has been obtained for layers thinner than 200 microm. The attenuation of the light inside the material by Beers law results in an attenuation of the index profile inside the material and in some cases the effective optical thickness of the material is lower than the physical thickness. This is an important and fundamental limitation in achieving high capacity holographic memories using photopolymers and cannot be analyzed using 2-D diffusion models. In this paper a model is proposed to describe the behavior of the photopolymers in 3-D. This model is applied to simulate the formation of profiles in depth for different photopolymer viscosities and different intensity attenuations inside the material.

Angular responses of the first and second diffracted orders in transmission diffraction grating recorded on photopolymer material.

Some of the theoretical models in the literature describing the mechanism of hologram formation in photopolymer materials predict the existence of higher harmonics in the Fourier expansion of the recorded refractive index. Nevertheless, quantitative information is only obtained for the first harmonic of the refractive index using Kogelniks Coupled Wave Theory. In this work we apply the Rigorous Coupled Wave Theory to demonstrate that when recording phase diffraction gratings in PVA/acrylamide photopolymer materials, a second order grating is also recorded in the hologram even when the material is exposed to a sinusoidal interference pattern. The influence of this second order grating on the efficiency of the first order for replay at the first on-Bragg angular replay condition is studied and the size of the 2nd harmonic examined.

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.

Comparison of peristrophic multiplexing and a combination of angular and peristrophic holographic multiplexing in a thick PVA/acrylamide photopolymer for data storage

Two different types of multiplexing are used to store 90 holograms at the same location in a polyvinyl alcoholacrylamide photopolymer material. In the first, the 90 holograms are stored using only peristrophic multiplexing, whereas in the second a combination of angular and peristrophic multiplexing is used. The results (diffraction efficiency and dynamic range, M#) obtained with these two multiplexing techniques are compared. With the first, the dynamic range was M# = 13 and with the second M# = 8. An exposure schedule method is used to calculate the exposure time necessary to store the holograms with a more uniform, higher diffraction efficiency.

First-harmonic diffusion-based model applied to a polyvinyl-alcohol– acrylamide-based photopolymer

The photopolymerization diffusion models give accurate comprehension of the mechanism of hologram formation inside photopolymer materials. Although several models have been proposed, these models share the common assumption that there is an interplay between the processes of monomer polymerization and monomer diffusion. Nevertheless, most of the studies to check the validity of the theoretical models have been done by using photopolymers of the DuPont™ type, or photopolymer materials with values of the monomer diffusion time similar to those of the DuPont material. We check the applicability of a modified diffusion-based model to a polyvinyl alcohol–acrylamide photopolymer. This material has the property of longer diffusion times for the monomer to travel from the unexposed to the exposed zones than in the case of other polymeric materials. Some interesting effects are observed and theoretically treated by using the modified first-harmonic diffusion-based model we propose.