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1 An Instructor s Solutions Manual to Accompany MECHANICAL VIBRATIONS: THEORY AND APPLICATIONS, 1 ST EDITION S. GRAHAM KELLY

2 2012 Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher except as may be permitted by the license terms below. ISBN-13: ISBN-10: Cengage Learning 200 First Stamford Place, Suite 400 Stamford, CT USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at: international.cengage.com/region. For product information and technology assistance, contact us at Cengage Learning Academic Resource Center, For permission to use material from this text or product, submit all requests online at Further permissions questions can be ed to permissionrequest@cengage.com. Cengage Learning products are represented in Canada by Nelson Education, Ltd. For your course and learning solutions, visit Purchase any of our products at your local college store or at our preferred online store NOTE: UNDER NO CIRCUMSTANCES MAY THIS MATERIAL OR ANY PORTION THEREOF BE SOLD, LICENSED, AUCTIONED, OR OTHERWISE REDISTRIBUTED EXCEPT AS MAY BE PERMITTED BY THE LICENSE TERMS HEREIN. READ IMPORTANT LICENSE INFORMATION Dear Professor or Other Supplement Recipient: Cengage Learning has provided you with this product (the Supplement ) for your review and, to the extent that you adopt the associated textbook for use in connection with your course (the Course ), you and your students who purchase the textbook may use the Supplement as described below. Cengage Learning has established these use limitations in response to concerns raised by authors, professors, and other users regarding the pedagogical problems stemming from unlimited distribution of Supplements. Cengage Learning hereby grants you a nontransferable license to use the Supplement in connection with the Course, subject to the following conditions. The Supplement is for your personal, noncommercial use only and may not be reproduced, posted electronically or distributed, except that portions of the Supplement may be provided to your students IN PRINT FORM ONLY in connection with your instruction of the Course, so long as such students are advised that they may not copy or distribute any portion of the Supplement to any third party. Test banks and other testing materials may be made available in the classroom and collected at the end of each class session, or posted electronically as described herein. Any material posted electronically must be through a password-protected site, with all copy and download functionality disabled, and accessible solely by your students who have purchased the associated textbook for the Course. You may not sell, license, auction, or otherwise redistribute the Supplement in any form. We ask that you take reasonable steps to protect the Supplement from unauthorized use, reproduction, or distribution. Your use of the Supplement indicates your acceptance of the conditions set forth in this Agreement. If you do not accept these conditions, you must return the Supplement unused within 30 days of receipt. All rights (including without limitation, copyrights, patents, and trade secrets) in the Supplement are and will remain the sole and exclusive property of Cengage Learning and/or its licensors. The Supplement is furnished by Cengage Learning on an as is basis without any warranties, express or implied. This Agreement will be governed by and construed pursuant to the laws of the State of New York, without regard to such State s conflict of law rules. Thank you for your assistance in helping to safeguard the integrity of the content contained in this Supplement. We trust you find the Supplement a useful teaching tool. Printed in the United States of America

3 INSTRUCTOR'S SOLUTIONS MANUAL TO ACCOMPANY MECHANICAL VIBRATIONS THEORY AND APPLICATIONS FIRST EDITION S. GRAHAM KELLY

4

5 Contents Preface vii Chapter 1 1 Chapter 2 55 Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter

6

7 Preface Mechanical vibrations is an applied engineering science. Students learn to apply previously learned sciences to realistic engineering problems. Students apply material learned in courses in statics, dynamics, mechanics of solids, fluid mechanics, calculus, differential equations and linear algebra to the solution of vibrations problems. The difference between a vibrations course and the aforementioned courses is the mathematical modeling aspect of vibrations. While studying vibrations students learn about mathematical modeling of systems with time as an independent variable. Students learn about assumptions made during the modeling process, coordinates and variables used, application of the basic laws of nature (including drawing of free-body diagrams), solving the mathematical problem, putting the solution in a form that can be used and most importantly how to interpret the solution to answer a given problem. Thus students learn the basic theory of mechanical vibrations and its application to problem solution and design. Mechanical vibrations is a precursor to engineering design. The purpose of an Instructor s Solution Manual is to provide solutions to end-ofchapter problems in the manner solved in the text. That is why this Solutions Manual is so lengthy. The solution to the vast majority of the problems is presented in such a way that is easy to follow. The end-of-chapter problems are broken into two types. Short answer questions are concept questions addressing the reader s understanding of basic concepts. They are broken into four subtypes: true/false, short answer, short calculation, and dimensions. The true/false questions ask the reader to evaluate a statement for its veracity and either explain why it is true or rewrite it to make the statement true. Short answer problems require the reader to formulate a short answer to a question while short calculation questions require the reader to make short calculations to test understanding of the concepts. The questions on dimensions are included for most chapters to review dimensions of basic quantities used in the chapter. The solutions to short answer questions are presented without the problem statement. Chapter problems involve lengthier calculations and may involve concepts form more than one chapter. The solutions of chapter problems are presented with a problem statement, a review of what is given in the problem statement with symbols assigned, a statement of what is desired in the solution, the solution of the problem and a statement regarding what the problem illustrates. The solution of the problem is presented in detail in most problems. This makes it easier for an instructor to decide what problems to assign. vii 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

8 The author acknowledges the help of his wife, Seala Fletcher-Kelly in preparing the figures and organizing the manual. However any mistakes in the solutions are solely mine. I would appreciate being informed of mistakes as you find them at sgraham@uakron.edu. That way I can post errata on the website for the book. S. Graham Kelly viii 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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