The classical problem of deflection of a cantilever beam of linear elastic material, under the action of an external vertical concentrated load at the free end, is analyzed.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 compared the theoretical results with the experimental results obtained in the laboratory.BELÉNDEZ, Tarsicio; NEIPP, Cristian; BELÉNDEZ, Augusto. "Large and small deflections of a cantilever beam". European Journal of Physics. Vol. 23, No. 3 (May 2002 1.-IntroductionIn this paper we shall analyze an example of a simple physical system, the deflections of a cantilever beam. We shall see that it is not complicated to formulate the equations governing its behaviour or to study it in a physics laboratory at university level. However, a differential equation with a non-linear term is also obtained. Moreover -as occurs with the simple pendulum for small oscillations- [1] when small deflections of the cantilever beam are considered, it is possible to find a simple analytical solution to the problem. In this sense, the study of large and small deflections of a cantilever beam presents a certain analogy with the study of large and small oscillations of a simple pendulum.The mathematical treatment of the equilibrium of cantilever beams does not involve a great difficulty [2][3][4]. Nevertheless, unless small deflections are considered, an analytical solution does not exist, since for large deflections a differential equation with a non-linear term must be solved. The problem is said to involve geometrical non-linearity [5,6]. An excellent treatment of the problem of deflection of a beam, built-it at one end and loaded at the other with a vertical concentrated force, can be found in "The Feynmann Lectures on Physics" [2], as well as in other university textbooks on physics, mechanics and elementary strength of materials. However, in these books the discussion is limited to the consideration of small deflections and they present a formula for the vertical deflection of the end free of the cantilever beam that shows a relation of proportionality between this deflection and the external force applied [2,4]. The analysis of large deflections of these types of cantilever beams of elastic material can be found in Landau's book on elasticity [5], and the solution in terms of elliptic integrals was obtained by Bisshopp and Drucker [7]. Nevertheless, the developments presented in these last references are difficult for first year university students.In this paper we analyze the problem of the deflection of a cantilever beam, in the ...

This paper deals with the non-linear oscillation of a simple pendulum and presents an approach for solving the non-linear differential equation that governs its movement by using the harmonic balance method. With this technique it is possible to easily obtain analytical approximate formulas for the period of the pendulum. As we shall see, these formulas show excellent agreement with the exact period calculated with the use of elliptical integrals, and they are valid for both small and large amplitudes of oscillation.The most significant feature of the treatment presented is its simplicity because for the level of approximation considered in this paper the required work can be done "by hand". KEY WORDS: Simple pendulum, large-angle period, harmonic balance method 3

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 0.75π (135• ) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency ω in terms of the complete elliptic integral of the first kind. We can conclude that for amplitudes lower than 135 o 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. Keywords: simple pendulum, large-angle period, angular displacement.Este artigo aborda a oscilação não-linear de um pêndulo simples e apresenta não apenas a fórmula exata do período mas também a dependencia temporal do deslocamento angular para amplitudes das oscilações e a freqüência angular para pequenas oscilações. O deslocamento angularé escrito em termos da função elíptica de Jacobi sn(u;m) usando as seguintes condições iniciais: o deslocamento angular inicialé diferente de zero enquanto que a velocidade angular inicialé zero. Os deslocamentos angulares são plotados usando Mathematica, um disponível programa simbólico de computador que nos permite plotar facilmente a função obtida. Como veremos, mesmo para amplitudes tão grandes quanto 0,75π (135 o )é possível usar a expressão para o deslocamento angular mas considerando a expressão exata para a freqüência angular w em termos da integral elíptica completa de primeira espécie. Concluímos que, para amplitudes menores que 135 o , o movimento periódico exibido por um pêndulo simplesé praticamente harmônico, mas suas oscilações não são isócronas (o períodoé uma função da amplitude inicial). Acreditamos que o presente estudo possa tornar-se um exercício conveniente e frutífero para o ensino e para uma melhor compreensão do pêndulo não-linear em cursos avançados de mecânica clássica na graduação. Palavras-chave: pêndulo simples, período a grandesângulos, deslocamento angular.Perhaps one of the nonlinear systems most studied and analyzed is the simple pendulum [1][2][3][4][5][6][7][8][9][10][11][12], which is the most popular textbook example of a nonlinear system and is studied not only in advanced but also in introductory university courses of...

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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.