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Single ConvolutionAir Springs

Belted Air Springs

Double ConvolutionAir Springs

Hans Dresig · Franz Holzweißig

Dynamics of Machinery

Theory and Applications

Translated by Dr. Wolf Grosskopf andAssoc. Prof. Dr.-Ing. Sven Esche

123

with CD-ROM

Univ.-Prof. Dr.-Ing. habil. Hans DresigMittelstr.1D-09244 Lichtenau OT AuerswaldeGermanyhdresig@web.deHomepage: http://www.dresig.de

Univ.-Prof. Dr.-Ing. habil. Franz HolzweißigKaitzer Str. 66D-01187 DresdenGermany

Translators

Dr. Wolf Grosskopf (Northridge)w.grosskopf@sbcglobal.net

Assoc. Prof. Dr.-Ing. Sven Esche(Stevens Institute of Technology)sesche@stevens.edu

ISBN 978-3-540-89939-6 e-ISBN 978-3-540-89940-2DOI 10.1007/978-3-540-89940-2Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2010929624

c© Springer-Verlag Berlin Heidelberg 2010This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevant protectivelaws and regulations and therefore free for general use.

Cover design: eStudio Calamar S.L.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Preface to the English Edition

This textbook, the translation of the ninth edition of the German book “Maschinen-dynamik” (published in its first edition in Germany and Austria in 1979), discussesdisciplines that in many countries are presented separately in lectures titled “Theoryof Mechanisms and Machines” (TMM) and “Vibrations” or “Theory of Vibrations”.The basic idea by my distinguished colleague, Prof. Holzweissig, was to combinethese disciplines into the “Dynamics of Machinery”. We worked out this concepttogether in the first three editions, relying on our experience gained in our collab-orations with engineers here in Saxony. Since he became a professor emeritus in1993, I have constantly updated the book since its 4th edition and have added moreexercises and sections.

This textbook is based on the four-semester lecture series on “Engineering Me-chanics” and confronts students directly with dynamic problems of their field. Dy-namics of machinery is viewed both as a typical field for the mathematical mod-eling of technological processes and as a branch of mechanical engineering thataddresses dynamic problems of power machines (reciprocating engines and turbo-machines), processing machines (printing machines, textile machines, packagingmachines), hoists and conveyors, agricultural machines and vehicles, as well as in-dustrial plants.

The book comprises relatively independent chapters that discuss typical issues ofthe dynamics of mechinery from the point of view of mechanical engineers. Chapter6 gives a general overview of linear oscillators and includes methods discussed inChaps. 3, 4 and 5. We were fully aware and accepted some overlap and repetitions,making many cross-references among sections. Chapter 7 discusses nonlinear andself-excited oscillators, for which calculations are becoming more and more relevantto practical application. Chapter 8 still does not contain any equations, but the rulescompiled there are important for engineering practice. We included a new chapter“on interrelations with system dynamics and mechatronics” to enhance the under-standing of these adjacent disciplines. This chapter prepares the readers for a moregeneral approach to solving problems of the dynamics of machinery by includingsensors and actuators.

V

VI Preface to the English Edition

The 60 problems with solutions are meant to help in grasping and consolidatingthe subjects taught. A new feature of this edition is the enclosed CD-ROM, whichdoes not only contain the student version of the SimulationX R© software, but alsoprogram parts that were used for solving the examples given in the book so that thereaders can work with them themselves.

The book was not only written for students, but also for practicing engineers.The examples from many fields of mechanical engineering and the specificationof parameter values, as well as references to guidelines and regulations, underlinethis. We have considered the way in which engineers think by discussing estimationmethods, rough calculations and minimal models, and by explaining many instruc-tive dynamic effects (gyroscopic effect, resonance, absorption, self-synchronization,. . . ) that are relevant for the design of machines that operate under a high dynamicload. The fast development of hardware and software entailed that nowadays thereis software available for almost every problem in the dynamics of machinery. Whilecomputers always provide some numbers and diagrams, the engineer is still respon-sible for the results. It remains the engineer’s job to provide the respective calcula-tion models, to assess the applicability of a software product, to check the result of asimulation, and to have some idea of the result to be expected before the calculationstarts.

Since the current development shows a trend towards refined modeling, wepointed out three aspects: the training of physical understanding, the utilization ofmodal analysis including sensitivity analysis, and the application of computer-aidedmethods. We stress the fact that the goal of calculations is not only to arrive at anumerical result, but to achieve a better understanding of the dynamic behavior ofreal objects and to be able to take design measures that are based in physical theory.

I would like to thank many specialists for their suggestions and tips, in partic-ular my former coworkers at the professorship of Dynamics of Machinery of theChemnitz University of Technology and my colleagues at ITI GmbH Dresden, inparticular Dipl.-Ing. Uwe Schreiber, who helped me work out practical examples.The whole book has been translated by Dr. Wolf Grosskopf (Northridge). I was verypleased that my former student, Prof. Dr.-Ing. Sven Esche (Stevens Institute of Tech-nology), agreed to proof-read the manuscript and to review the terminology. I am inparticular grateful for this. I would like to give special praise to the committed workof Dipl.-Ing. Andreas Abel who prepared the final manuscript and thereby took intoaccount all my many extra wishes, so that the manuscript could be submitted to thepublisher ready for printing.

Hans DresigAuerswalde, in May 2010

Contents

Purpose and Structure of the Dynamics of Machinery . . . . . . . . . . . . . . . . . . 1

1 Model Generation and Parameter Identification . . . . . . . . . . . . . . . . . . . 51.1 Classification of Calculation Models . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Determination of Mass Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.2 Mass and Position of the Center of Gravity . . . . . . . . . . . . . . . 151.2.3 Moment of Inertia about an Axis . . . . . . . . . . . . . . . . . . . . . . . 171.2.4 Moment of Inertia Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3 Spring Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.3.1 General Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.3.2 Machine Elements, Sub-Assemblies . . . . . . . . . . . . . . . . . . . . 291.3.3 Rubber Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.3.4 Problems P1.1 to P1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.3.5 Solutions S1.1 to S1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.4 Damping Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.4.1 General Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.4.2 Methods for Determining Characteristic Damping

Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.4.3 Empirical Damping Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.5 Characteristic Excitation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 551.5.1 Periodic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551.5.2 Transient Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561.5.3 Problems P1.4 to P1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621.5.4 Solutions S1.4 to S1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2 Dynamics of Rigid Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.2 Kinematics of a Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

VII

VIII Contents

2.2.1 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.2.2 Kinematic Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.2.3 Kinematics of the Gimbal-Mounted Gyroscope . . . . . . . . . . . 752.2.4 Problems P2.1 and P2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762.2.5 Solutions S2.1 and S2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.3 Kinetics of the Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.3.1 Kinetic Energy and Moment of Inertia Tensor . . . . . . . . . . . . 822.3.2 Principles of Linear Momentum and of Angular Momentum 872.3.3 Kinetics of Edge Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.3.4 Problems P2.3 and P2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932.3.5 Solutions S2.3 and S2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

2.4 Kinetics of Multibody Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.4.1 Mechanisms with Multiple Drives . . . . . . . . . . . . . . . . . . . . . . 1002.4.2 Planar Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122.4.3 States of Motion of a Rigid Machine . . . . . . . . . . . . . . . . . . . . 1222.4.4 Solution of the Equations of Motion . . . . . . . . . . . . . . . . . . . . 1242.4.5 Example: Press Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1292.4.6 Problems P2.5 to P2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1332.4.7 Solutions S2.5 to S2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

2.5 Joint Forces and Foundation Loading . . . . . . . . . . . . . . . . . . . . . . . . . . 1412.5.1 General Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1412.5.2 Calculating Joint Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1422.5.3 Calculation of the Forces Acting onto the Frame . . . . . . . . . . 1452.5.4 Joint Forces in the Linkage of a Processing Machine . . . . . . . 1482.5.5 Problems P2.9 and P2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1502.5.6 Solutions S2.9 and S2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

2.6 Methods of Mass Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1532.6.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1532.6.2 Counterbalancing of Rigid Rotors . . . . . . . . . . . . . . . . . . . . . . 1532.6.3 Mass Balancing of Planar Mechanisms . . . . . . . . . . . . . . . . . . 1602.6.4 Problems P2.11 to P2.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1672.6.5 Solutions S2.11 to S2.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

3 Foundation and Vibration Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1773.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1773.2 Foundation Loading for Periodic Excitation . . . . . . . . . . . . . . . . . . . . 181

3.2.1 Minimal Models with One Degree of Freedom . . . . . . . . . . . . 1813.2.2 Block Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1913.2.3 Foundations with Two Degrees of Freedom – Vibration

Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2003.2.4 Example: Vibrations of an Engine-Generator System . . . . . . 2043.2.5 Problems P3.1 to P3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2063.2.6 Solutions to Problems S3.1 to S3.3 . . . . . . . . . . . . . . . . . . . . . 208

3.3 Foundations under Impact Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2113.3.1 Modeling Forging Hammers . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Contents IX

3.3.2 Calculation Model with Two Degrees of Freedom . . . . . . . . . 2133.3.3 Problems P3.4 to P3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2163.3.4 Solutions S3.4 to S3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

4 Torsional Oscillators and Longitudinal Oscillators . . . . . . . . . . . . . . . . . 2234.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2234.2 Free Vibrations of Torsional Oscillators . . . . . . . . . . . . . . . . . . . . . . . . 229

4.2.1 Models with Two Degrees of Freedom . . . . . . . . . . . . . . . . . . 2294.2.2 Oscillator Chains with Multiple Degrees of Freedom . . . . . . 2344.2.3 Evaluation of Natural Frequencies and Mode Shapes . . . . . . 2384.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2424.2.5 Problems P4.1 to P4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2514.2.6 Solutions S4.1 to S4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

4.3 Forced Vibrations of Discrete Torsional Oscillators . . . . . . . . . . . . . . 2594.3.1 Periodic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2594.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2634.3.3 Transient Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2724.3.4 Problems P4.4 to P4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2784.3.5 Solutions S4.4 to S4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

4.4 Absorbers and Dampers in Drive Systems . . . . . . . . . . . . . . . . . . . . . . 2834.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2834.4.2 Design of an Undamped Absorber . . . . . . . . . . . . . . . . . . . . . . 2844.4.3 Design of a Spring-Constrained Damper . . . . . . . . . . . . . . . . . 2864.4.4 Design of a Springless Damper . . . . . . . . . . . . . . . . . . . . . . . . . 2894.4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

4.5 Parameter-Excited Vibrations by Gear Mechanisms with VaryingTransmission Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2954.5.1 Problem Formulation/Equation of Motion . . . . . . . . . . . . . . . . 2954.5.2 Solution of the Equation of Motion, Stability Behavior . . . . . 2974.5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2994.5.4 Problems P4.7 and P4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3064.5.5 Solutions S4.7 and S4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

5 Bending Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115.1 Problem Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115.2 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

5.2.1 Self-Centering in a Symmetrical Rotor . . . . . . . . . . . . . . . . . . 3125.2.2 Passing through the Resonance Point . . . . . . . . . . . . . . . . . . . . 3155.2.3 Rotating Shaft with Disk (Gyroscopic Effect) . . . . . . . . . . . . . 3165.2.4 Bending Oscillators with a Finite Number of Degrees of

Freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3265.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3285.2.6 Problems P5.1 to P5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3355.2.7 Solutions S5.1 to S5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

5.3 Beam with Distributed Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

X Contents

5.3.1 General Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3385.3.2 Straight Beam on Two Supports . . . . . . . . . . . . . . . . . . . . . . . . 3435.3.3 Estimates by Dunkerley and Neuber . . . . . . . . . . . . . . . . . . . . 346

5.4 Model Generation for Rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3475.4.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3475.4.2 Example: Grinding Spindle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

5.5 Problems P5.4 to P5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3515.6 Solutions S5.4 to S5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

6 Linear Oscillators with Multiple Degrees of Freedom . . . . . . . . . . . . . . 3556.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3556.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

6.2.1 Mass, Spring, and Compliance matrix . . . . . . . . . . . . . . . . . . . 3586.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3636.2.3 Problems P6.1 to P6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3726.2.4 Solutions S6.1 to S6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

6.3 Free Undamped Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3756.3.1 Natural Frequencies, Mode Shapes, Eigenforces . . . . . . . . . . 3756.3.2 Orthogonality and Modal Coordinates . . . . . . . . . . . . . . . . . . . 3786.3.3 Initial Conditions, Initial Energy, Estimates . . . . . . . . . . . . . . 3816.3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3836.3.5 Problems P6.4 to P6.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3956.3.6 Solutions S6.4 to S6.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

6.4 Structure and Parameter Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3996.4.1 Rayleigh Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3996.4.2 Sensitivity of Natural Frequencies and Mode Shapes . . . . . . . 4006.4.3 Reduction of Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . 4056.4.4 Influence of Constraints on Natural Frequencies and Mode

Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4076.4.5 Examples of the Reduction of Degrees of Freedom . . . . . . . . 4116.4.6 Problems P6.7 to P6.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4196.4.7 Solutions S6.7 to S6.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

6.5 Forced Undamped Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4266.5.1 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4266.5.2 Harmonic Excitation (resonance, absorption) . . . . . . . . . . . . . 4276.5.3 Transient Excitation (Rectangular Impulse) . . . . . . . . . . . . . . 4336.5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4376.5.5 Problems P6.10 to P6.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4406.5.6 Solutions S6.10 to S6.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

6.6 Damped Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4446.6.1 Determination of Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4446.6.2 Free Damped Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4456.6.3 Harmonic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4476.6.4 Periodic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4526.6.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

Contents XI

6.6.6 Problems P6.13 to P6.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4606.6.7 Solutions S6.13 to S6.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460

7 Simple Nonlinear and Self-Excited Oscillators . . . . . . . . . . . . . . . . . . . . . 4657.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4657.2 Nonlinear Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

7.2.1 Undamped Free Nonlinear Oscillators . . . . . . . . . . . . . . . . . . . 4687.2.2 Forced Vibrations with Harmonic Excitation . . . . . . . . . . . . . 4717.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4767.2.4 Problems P7.1 to P7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4887.2.5 Solutions S7.1 and S7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

7.3 Self-Excited Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4907.3.1 General Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4907.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4917.3.3 Problems P7.3 and P7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5007.3.4 Solutions S7.3 and S7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

8 Rules for Dynamically Favorable Designs . . . . . . . . . . . . . . . . . . . . . . . . . 505

9 Relations to System Dynamics and Mechatronics . . . . . . . . . . . . . . . . . . 5119.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5119.2 Closed-Loop Controlled Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

9.2.1 General Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5149.2.2 Example: Influencing Frame Vibrations by a Controller . . . . 516

Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

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Purpose and Structure of theDynamics of Machinery

The purpose of the dynamics of machinery (aka machine dynamics) is to applythe knowledge from the field of dynamics to specific problems in engineering. Itsdevelopment is closely linked to developments in mechanical engineering.

Dynamic problems first occurred in power and work machines. Torsional vibra-tions were observed in reciprocating engines, and bending vibrations put turbinecomponents at risk. Explaining such phenomena has long been the only task of thedynamics of machinery, as reflected in the standard publications, e. g. [1].

Knowledge of the dynamics of machinery is also required for designing machinesthat are based on dynamic principles. This includes hammers, robots, stemmers,oscillating conveyors, screens, vibrators, textile mandrils, centrifuges, etc.

Dynamic problems stepped into the foreground with the ever-increasing oper-ating speed and the enforcement of lightweight design principles in all fields ofprocessing machines, agricultural machines, machine tools, printing machines, andconveyors. To manage their multiplicity, it became necessary to define and answerthe basic questions, while abstracting as much as possible from any specific ma-chine. In this way, dynamics of machinery grew into an independent discipline thatis part of the tool set of each mechanical engineer.

While 50 years ago people thought that dealing with vibration problems willremain the domain of only a few specialists, today we expect a large number ofengineers to have a precise understanding of the dynamic processes in a machine.High-performance machines are to be sized not just based on static, but often pri-marily based on dynamic considerations. Thus, the use of calculation methods ofstructural durability depends on the certainty of the load assumptions that resultfrom a machine dynamics calculation. An engineer has to know the physical lawsthat have an influence on periodic permanent loads, impacts, starting and stoppingprocesses in a machine.

The way in which engineers work has to be taken into account, though. It isgoverned by the requirement to solve a practical problem in a short time and ateconomically justifiable expenditure. Frequently, fast decisions have to be made sothat waiting for the scientific clarification of detailed questions is not an option.Instead, an engineer has to consider all available information and adjust the solution

1

2 Purpose and Structure of the Dynamics of Machinery

of the problem to the existing state of the art. An important skill that an engineershould have is to be able to apply an imperfect or incomplete theory as long as thereis no better one available. This requires, of course, a wealth of knowledge that willnot always include the entire train of thought that leads to an equation or computerprogram. What is important, however, is to know their scope and make use of waysto check the results obtained using rough estimates.

The diagram below, in Fig. 0.1, outlines the path that is followed when solving adesign problem. It is desirable to have someone who is able to go down the externalpath, that is, someone who glances at the machine (or listens to it) and knows whathas to be changed to obtain the desired effect. Chapter 8 provides a few generalrules that an experienced designer can rely on. However, they are incomplete andknowing them is not sufficient to solve all problems. Simply put, what matters aredecisions to change something in the structure or parameters of a specific object insuch a way that the dynamic performance of that object is improved.

Since this selection of decisions is anything but trivial, a step-by-step approachas indicated on the left-hand side of the diagram is preferred. The essential steps in-clude the identification of a mechanical problem in a design (or technological) task,and the formulation of a mathematical problem from it (which is part of modeling).One has to understand the physical behavior before generating a model. Chapter 1discusses issues of model generation and parameter identification.

In the course of the past two decades, computer-aided analysis using commercialsoftware has become the standard in the dynamics of machinery as well. Today’sengineer is spared the many efforts that the solution of the mathematical problemsused to involve. In recent years, the interpretation of the mathematical solution andthe implementation of mechanical principles in a design solution have become in-creasingly important engineering tasks. Today, dynamic phenomena can be bettercalculated in advance and taken advantage of, e. g. dynamic balancing, the effects ofgyroscopes, absorbers, dampers, and nonlinear effects (e. g. self-synchronization).

Design problemProblem definition

Mechanical problemModel generation Problem

Mathematical problemSoftware application Experience

Mathematical solutionInterpretation Rules

Mechanical solutionImplementation

Design solution

Fig. 0.1 Tasks in dynamics of machinery

Figure 0.2 shows how the field of dynamics of machinery relates to other disciplines.Rotor dynamics [8], or vehicle dynamics [27], could be related to these disciplinesin a similar way, and they overlap with the dynamics of machinery. Unlike rotor

Purpose and Structure of the Dynamics of Machinery 3

and vehicle dynamics, machine dynamics deals with many different objects andproblems, from machine elements [22] to complex designs where close links tostructural dynamics (and not just with regard to foundations) exist [19].

The theoretical foundation of the dynamics of machinery is formed by mathe-matics and physics [32], including, of course, almost all fields of engineering me-chanics [11], [30], and in particular, the theory of vibrations [6], [10], [16], [24].The findings of machine dynamics studies influence the design and sizing of realmachines, and there are close links to design theory, machine elements, structuralstability, drive engineering, and acoustics of machines. The most closely related dis-ciplines are drive engineering (for the drives) [4], [9], [21] and structural dynamics(for the frames) [19], [25] if, from a design perspective, one views a machine as acombination of a drive and a support system.

Technical mechanics

Structural dynamics (FEM, MKS) System dynamics

Structural dynamics

Design theory (CAD) Acoustics of machines

Structural durability

Dynamics of Machinery

Vibration measurement technology

Machine elements Information technology

Drive engineering Electrical engineering

Theory of vibrations

Fig. 0.2 Links of the dynamics of machinery to other disciplines

The dynamics of machinery always involves real objects that are subjected to loadsthat vary over time, and that are exposed to, or take advantage of, the effects ofinertia. Frequently, the purpose is also to avoid malfunctions or damages. The find-ings of the theory of vibrations provide the basis for understanding many real-worldphenomena. The technological development of many machines has influenced thetheory of vibrations in that it continuously provided new practical questions thatcould not be answered using the body of knowledge and theories available.

Problems of modeling in the dynamics of machinery are closely related to sys-tem dynamics [2], [3], [14], [18], and vibration measurement technology, but alsoinclude methods for evaluating and assessing [15], [17] vibration phenomena. Thedynamics of machinery utilizes the findings in fields such as structural dynamics [7]and multibody dynamics [3], [28] that are often found – implemented using numer-ical mathematics – in commercial software products. Quantitative improvements ofthe calculation models are not just achieved by mutual perfection of computer andmeasurement technology. There are crosslinks with electrical engineering when itcomes to electric drives and applying findings of control engineering and computer

4 Purpose and Structure of the Dynamics of Machinery

science. Unlike mechatronics, the main task in the dynamics of machinery, however,is the control of inertia forces.

A special field of the dynamics of machinery involves the collection and classifi-cation of proven computational models and their associated characteristics that arerequired as input data for computer programs. Advanced and proven computationalmodels of real machines are often kept confidential by companies. They representthe outcome of a long development process and are very detailed and powerful. Ifrequested (e. g. for calculating specific machine elements), they are often accessibleto third parties.

Students and engineers in the field have to understand the computational mod-els commercial software products are based on and know the physical background.Everyone should also have some idea of the scales of masses, spring constants,damping coefficients, forces, frequencies, etc. that they have to deal with. There-fore, this book uses real-world parameter values for most of the problems presented.One should always try to anticipate the results that the computer will provide. Withsome practice, an engineer will often be able to predict the order of magnitude ofreal parameter values and results of a calculation.

For clarity reasons, the material discussed in this textbook has been structuredfrom simple to complex concepts. Basic knowledge of engineering mechanics isrequired. Chapter 2, which deals with the dynamics of rigid bodies, probably alsodiscusses some known phenomena. Understanding the balance of forces in a rigid-body system, however, provides the basis for understanding many other dynamicphenomena. Specific oscillation systems are discussed in Chaps. 3 to 5, but thesechapters focus on ”‘classical”’ fields of the dynamics of machinery (foundation de-sign, vibration isolation, torsional and translational vibrations, bending vibrations)and their major applications. Chapter 6 repeats the material presented in Chaps. 3 to5 at a higher level and includes the theory of linear vibrations, a field that can be ap-plied to many machines. Chapter 7 gives an insight into some nonlinear phenomenaand the behavior of self-excited oscillators that are relevant for mechanical engi-neering. The rules provided in Chap. 8 refer to the qualitative side of the dynamicsof machinery.