Large and small deflections of a cantilever beam

"Large and small deflections of a cantilever beam".Large and small deflections of a cantilever beam". ABSTRACT 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.Large and small deflections of a cantilever beam". In 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 [2-4]. 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 …