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Introduction of modern problems into beginning mechanics curricula Abstract

Nowadays, in the context of smart materials, spatially varying material properties (such as occurs in functionally graded materials) are being investigated. Also, structures with varying cross- sectional areas have been studied with a view towards shape optimization. Up to now such problems have not been introduced into beginning and intermediate mechanics courses because they involve differential equations with variable coefficients which typically do not have analytic solutions. Also non-linear effects are precluded because of the lack of analytical solutions. However these problems can readily be handled using numerical ODE solvers, such as in MAPLE®. Students exposure to such problems could considerably enrich their knowledge and understanding. Moreover, the process aids in the goal of integrating computation throughout the curriculum. Here two classes of statics problems are presented, namely: (i) the effect of Young’s Modulus variation on the end deflection of an axially loaded rod and (ii) of all the rod shapes with exponentially varying cross-sectional areas, with all rods having the same volume, which one leads to the minimum end deflection under axial load. A more traditional case dealing with dynamics is also presented, namely: the solution of a non-linear problem involving the effects of friction on the velocity and reactions on a bead sliding on a rough circular vertical track.

Introduction

This work is a third in a series [1], [2] aimed at extending basic knowledge, and improving understanding, in introductory mechanical courses. Moreover, it aids in an ABET goal of integrating computer usage throughout the curricula. Several problems dealing with strength of materials are discussed. The first involves spatially varying material properties (such as occurs in functionally graded materials (FGM) [3]). The effect of Young’s Modulus variation on the end deflection of an axially loaded rod is given. The second problem involves structures with varying cross-sectional areas with a view towards shape optimization. The question posed is: of all the rod shapes with exponentially varying cross-sectional areas, with all rods having the same volume, which one leads to the minimum end deflection when the rod is axially loaded. Up to now such problems have not been introduced into introductory mechanics courses because they involve differential equations with variable coefficients which typically do not have analytic solutions. However these problems can readily be handled using a finite difference scheme such as in MAPLE® (MATLAB® or other packages could also be used). The students should be aware of the nature of finite difference schemes. A simple illustrative example is given in reference [1]. A final study deals with the solution of a dynamics non-linear problem involving a bead sliding on a rough vertical track. Non-linear problems typically do not have analytic solutions and numerical methods are used. The effects of friction on the velocity and normal reactions are given.

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Mazzei, A., & Scott, R. (2008, June), Introduction Of Modern Problems Into Beginning Mechanics Curricula Paper presented at 2008 Annual Conference & Exposition, Pittsburgh, Pennsylvania. 10.18260/1-2--3111