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# Statics with MATLAB Dan B. Marghitu Mihai Dupac Nels H. Madsen

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Statics with MATLAB® Dan B. Marghitu · Mihai Dupac Nels H. Madsen Statics with MATLAB� Dan B. Marghitu • Mihai Dupac Nels H. Madsen Statics with MATLAB� 123 Dan B. Marghitu Mechanical Engineering Auburn University Auburn, AL USA Nels H. Madsen Samuel Ginn College of Engineering Auburn University Auburn, AL USA Mihai Dupac Talbot Campus, School of Design, Engineering and Computing Bournemouth University Poole UK ISBN 978-1-4471-5109-8 ISBN 978-1-4471-5110-4 (eBook) DOI 10.1007/978-1-4471-5110-4 Springer London Heidelberg New York Dordrecht Library of Congress Control Number: 2013936003 � Springer-Verlag London 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface Engineering mechanics involves the development of mathematical models of the physical world. Statics, a branch of mechanics, addresses the forces acting on and in mechanical objects and systems in equilibrium, and the impact those forces have on the motion, or lack thereof, of those systems. The project deals with the understanding of the mechanical behavior of complex engineering structures and components. The tools of formulating the mathematical equations and the solution methods are discussed. An understanding of forces in and equilibrium of structures and components is most important for their design. MATLAB is a modern tool that has transformed mathematical methods, because MATLAB not only provides numerical calculations but also facilitates analytical or symbolic calculations using the computer. The present project uses MATLAB as a tool to solve problems. The intent is to show the convenience of MATLAB for theory and applications in statics. This approach will significantly enhance the student’s ability to use MATLAB both within statics and beyond. Using examples of problems the MATLAB syntax will be demonstrated. MATLAB is very useful in the process of deriving solutions for any problem in statics. The project will include a large number of problems that are solved using MATLAB. Specific functions dealing with statics topics are introduced and cre- ated. The programs will be available on a website accompanying the project. The main distinction of the study from other projects and books is the use of symbolic MATLAB for both theory and applications. Special attention is given to the solutions of the problems that are solved analytically and numerically using MATLAB. The figures generated with MATLAB will reinforce visual learning for students as they study the programs. This project is intended primarily for use in a one semester course in statics and could be used in a two semester sequence of courses in statics and dynamics. The project can be used for classroom instruction, for self-study, and in a distance learning environment. It would be appropriate for use as a text at the undergrad- uate level. Chapter 1 is intended to give an introduction to vector mechanics. The reason for this chapter is that many scientific concepts used to describe the physical world, have attributes not only of size or magnitude, but also have associated with them the idea of a direction. Examples of such quantities include force, moment, v and couple. This chapter provides a starting point for students wishing to develop the basic principle of mechanics. MATLAB is used to calculate the magnitudes of vectors, direction cosines, dot products, cross products, scalar triple products, vector triple products, and derivatives of vector functions. The examples presented begin with a symbolic development, followed by numerical evaluation and the generation of vector figures, all done within MATLAB. Chapter 2 demonstrates the use of MATLAB in finding the moment of a vector about a point, the moment of a system of vectors, the moment of a couple about a point, the equivalence of systems of vectors, and the force vector and the moment of a force. The figures are depicted using graphical functions built in MATLAB. This chapter also provides an introduction to the basic principles of mechanics. Chapter 3, centroids and center of mass, presents the principles and details of centroids (also known as the geometric center and connected to the first moment of area) and surface properties, their meaning and importance. All the presentation will be detailed (centroid of a set of points, centroid of a curve, surface or solid, Guldinus-Pappus theorems, parallel-axis theorem) and in some cases followed by examples using MATLAB. External functions can be introduced to calculate the centroids of complex figures. The concepts of the first moment are also useful in analyzing distributed forces. Chapter 4 analyzes many of the equilibrium problems that are encountered in engineering applications. The equilibrium equations are stated and various types of supports are depicted. The unknown forces and moments acting on bodies are communicated using free-body diagrams and the equilibrium equations are determined. If an object is in equilibrium, the net moment about any point due to the forces and couples acting on the object is zero and the sum of the forces must also be zero. The calculation of moments is explained and the concept of equiv- alent systems of forces and moments is introduced. In engineering, the term structure can refer to any object that has the capacity to support and exert loads. This chapter studies structures composed of interconnected parts or links. The forces and couples acting on the structure as a total as well as on its individual members are determined. Trusses, which are composed of two-force members, are studied and then frames and machines are considered. MATLAB functions are applied to find and solve the algebraic static equations. The objective of Chap. 5 is to provide an introduction to friction. Friction forces in engineering applications, have important effects both desirable and undesirable. The Coulomb law of friction is used to find the maximum friction forces that can be exerted by contacting surfaces and the friction forces exerted by sliding sur- faces. Threaded connections are also analyzed.