# Vibration-problems-in-engineering by Timoshenko

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<ul><li> VIBRATION PROBLEMS IN ENGINEERING BY S. TIMOSHENKO Professor of Theoretical and Engineering Mechanics Stanford University SECOND EDITIONFIFTH PRINTING NEW YORK D. VAN NOSTRAND COMPANY, INC. 250 FOURTH AVENUE </li> <li> COPYRIGHT, 1928, 1937, BY D. VAN NOSTRAND COMPANY, INC. All Rights Reserved This book or any part thereof may not be reproduced in any form without written permission from the publisher,, First Published . . . October, 1928 Second Edition . . July, 1937 RcpruiUd, AuyuU, 1^41, July, UL't J, Auyu^t, t'^44, A/ PRINTED IN THE USA </li> <li> PREFACE TO THE SECOND EDITION In the preparation of the manuscript for the second edition of the book, the author's desire was not only to bring the book up to date by including some new material but also to make it more suitable for teaching purposes. With this in view, the first part of the book was entirely re- written and considerably enlarged. A number of examples and problems with solutions or with answers were included, and in many places new material was added. The principal additions are as follows : In the first chapter a discussion of forced vibration with damping not proportional to velocity is included, and an article on self-excited vibration. In the chapter on non-linear sys- tems an article on the method of successive approximations is added and it is shown how the method can be used in discussing free and forced vibra- tions of systems with non-linear characteristics. The third chapter is made more complete by including in it a general discussion of the equation of vibratory motion of systems with variable spring characteristics. The fourth chapter, dealing with systems having several degrees of freedom, is also Considerably enlarged by adding a general discussion of systems with viscous damping; an article on stability of motion with an application in studying vibration of a governor of a steam engine; an article on whirling of a rotating shaft due to hysteresis; and an article on the theory of damp- ing vibration absorbers. There are also several additions in the chapter on torsional and lateral vibrations of shafts. The author takes this opportunity to thank his friends who assisted in various ways in the preparation of the manuscript* and particularly Professor L. S. Jacobsen, who read over the complete manuscript and made many valuable suggestions, and Dr. J. A. Wojtaszak, who checked prob- lems of the first chapter. STEPHEN TIMOSHENKO STANFORD UNIVERSITY, May 29, 1937 </li> <li> PREFACE TO THE FIRST EDITION With the increase of size and velocity in modern machines, the analysis of vibration problems becomes more and more important in mechanical engineering design. It is well known that problems of great practical significance, such as the balancing of machines, the torsional vibration of shafts and of geared systems, the vibrations of turbine blades and turbine discs, the whirling of rotating shafts, the vibrations of railway track and bridges under the action of rolling loads, the vibration of foundations, can be thoroughly understood only on the basis of the theory of vibration. Only by using this theory can the most favorable design proportions be found which will remove the working conditions of the machine as far as possible from the critical conditions at which heavy vibrations may occur. In the present book, the fundamentals of the theory of vibration are developed, and their application to the solution of technical problems is illustrated by various examples, taken, in many cases, from actual experience with vibration of machines and structures in service. In developing this book, the author has followed the lectures on vibration given by him to the mechanical engineers of the Westinghouse Electric and Manufacturing Company during the year 1925, and also certain chapters of his previously published book on the theory of elasticity.* The contents of the book in general are as follows: The first chapter is devoted to the discussion of harmonic vibrations of systems with one degree of freedom. The general theory of free and forced vibration is discussed, and the application of this theory to balancing machines and vibration-recording instruments is shown. The Rayleigh approximate method of investigating vibrations of more com- plicated systems is also discussed, and is applied to the calculation of the whirling speeds of rotating shafts of variable cross-section. Chapter two contains the theory of the non-harmonic vibration of sys- tems with one degree of freedom. The approximate methods for investi- gating the free and forced vibrations of such systems are discussed. A particular case in which the flexibility of the system varies with the time is considered in detail, and the results of this theory are applied to the inves- tigation of vibrations in electric locomotives with side-rod drive. * Theory of Elasticity, Vol. II (1916) St. Petersburg, Russia. v </li> <li> vi PREFACE TO THE FIRST EDITION In chapter three, systems with several degrees of freedom are con- sidered. The general theory of vibration of such systems is developed, and also its application in the solution of such engineering problems as: the vibration of vehicles, the torsional vibration of shafts, whirling speeds of shafts on several supports, and vibration absorbers. Chapter four contains the theory of vibration of elastic bodies. The problems considered are : the longitudinal, torsional, and lateral vibrations of prismatical bars; the vibration of bars of variable cross-section; the vibrations of bridges, turbine blades, and ship hulls; the theory of vibra- tion of circular rings, membranes, plates, and turbine discs. Brief descriptions of the most important vibration-recording instru- ments which are of use in the experimental investigation of vibration are given in the appendix. The author owes a very large debt of gratitude to the management of the Westinghouse Electric and Manufacturing Company, which company made it possible for him to spend a considerable amount of time in the preparation of the manuscript and to use as examples various actual cases of vibration in machines which were investigated by the company's engineers. He takes this opportunity to thank, also, the numerous friends who have assisted him in various ways in the preparation of the manuscript, particularly Messr. J. M. Lessells, J. Ormondroyd, and J. P. Den Hartog, who have read over the complete manuscript and have made many valuable suggestions. He is indebted, also, to Mr. F. C. Wilharm for the preparation of draw- ings, and to the Van Nostrand Company for their care in the publication oi the book. S. TIMOSHENKO ANN ARBOR, MICHIGAN, May 22, 1928. </li> <li> CONTENTS CHAPTER I PAGE HARMONIC VIBRATIONS OF SYSTEMS HAVINO ONE DEGREE OF FREEDOM 1 . Free Harmonic Vibrations 1 2. Torsional Vibration 4 3. Forced Vibrations 8 4. Instruments for Investigating Vibrations 19 5. Spring Mounting of Machines 24 6. Other Technical Applications 26 /V. Damping 30 ^78. Free Vibration with Viscous Damping 32 9. Forced Vibrations with Viscous Damping 38 10. Spring Mounting of Machines with Damping Considered 51 11. Free Vibrations with Coulomb's Damping 54 12. Forced Vibrations with Coulomb's Damping and other Kinds of Damping. 57 13. Machines for Balancing 62 14. General Case of Disturbing Force 64 v/15. Application of Equation of Energy in Vibration Problems 74 16. Rayleigh Method 83 17. Critical Speed of a Rotating Shaft 92 18. General Case of Disturbing Force 98 19. Effect of Low Spots on Deflection of Rails 107 20. Self-Excited Vibration 110 CHAPTER IT VIBRATION OF SYSTEMS WITH NON-LINEAR CHARACTERISTICS 21 . Examples of Non-Linear Systems 114 22. Vibrations of Systems with Non-linear Restoring Force 119 23. Graphical Solution 121 24. Numerical Solution 126 25. Method of Successive Approximations Applied to Free Vibrations 129 26. Forced Non-Linear Vibrations 137 CHAPTER 111 SYSTEMS WITH VARIABLE SPRING CHARACTERISTICS 27. Examples of Variable Spring Characteristics 151 28. Discussion of the Equation of Vibratory Motion with Variable Spring Characteristics 160 29. Vibrations in the Side Rod Drive System of Electric Locomotives 167 vii </li> <li> via CONTENTS CHAPTER IV PAGE SYSTEMS HAVING SEVERAL DEGREES OF FREEDOM 30. d'Alembert's Principle and the Principle of Virtual Displacements 182 31. Generalized Coordinates and Generalized Forces 185 32. Lagrange's Equations 189 33. The Spherical Pendulum 192 34. Free Vibrations. General Discussion 194 35. Particular Cases 206 36. Forced Vibrations 208 37. Vibration with Viscous Damping 213 38. Stability of Motion 216 39. Whirling of a Rotating Shaft Caused by Hysteresis 222 40. Vibration of Vehicles 229 41. Damping Vibration Absorber 240 CHAPTER V TORSIONAL AND LATERAL VIBRATION OF SHAFTS 42. Free Torsional Vibrations of Shafts 253 43. Approximate Methods of Calculating Frequencies of Natural Vibrations . . 258 44. Forced Torsional Vibration of a Shaft with Several Discs 265 45. Torsional Vibration of Diesel Engine Crankshafts 270 46. Damper with Solid Friction 274 47. Lateral Vibrations of Shafts on Many Supports 277 48. Gyroscopic Effects on the Critical Speeds of Rotating Shafts 290 49. Effect of Weight of Shaft and Discs on the Critical Speed 299 50. Effect of Flexibility of Shafts on the Balancing of Machines 303 CHAPTER VI VIBRATION OF ELASTIC BODIES 51. Longitudinal Vibrations of Prismatical Bars 307 52. Vibration of a Bar with a Load at the End 317 53. Torsional Vibration of Circular Shafts 325 54. Lateral Vibration of Prismatical Bars 331 55. The Effect of Shearing Force and Rotatory Inertia 337 V56. Free Vibration of a Bar with Hinged Ends 338 Jbl. Other End Fastenings 342 >/58. Forced Vibration of a Beam with Supported Ends 348 59. Vibration of Bridges 358 60. Effect of Axial Forces on Lateral Vibrations 364 61. Vibration of Beams on Elastic Foundation 368 62. Ritz Method 370 63. Vibration of Bars of Variable Cross Section 376 64. Vibration of Turbine Blades . . . . . . 382 </li> <li> CONTENTS ix PAGE 65. Vibration of Hulls of Ships 388 06. Lateral Impact of Bars 392 67. Longitudinal Impact of Prismatical Bars 397 *8. Vibration of a Circular Ring 405 '69. Vibration of Membranes 411 70. Vibration of Plates 421 71. Vibration of Turbine Discs 435 APPENDIX VIBRATION MEASURING INSTRUMENTS 1. General 443 2. Frequency Measuring Instruments 443 3. The Measurement of Amplitudes 444 4. Seismic Vibrographs 448 5. Torsiograph 452 6. Torsion Meters 453 7. Strain Recorders 457 AUTHOR INDEX 463 SUBJECT INDEX 467 </li> <li> CHAPTER I HARMONIC VIBRATIONS OF SYSTEMS HAVING ONE DEGREE OF FREEDOM 1. Free Harmonic Vibrations. If an elastic system, such as a loaded beam, a twisted shaft or a deformed spring, be disturbed from its position of equilibrium by an impact or by the sudden application and removal of an additional force, the elastic forces of the member in the disturbed posi- tion will no longer be in equilibrium with the loading, and vibrations will ensue. Generally an elastic system can perform vibrations of different modes. For instance, a string or a beam while vibrating may assume the different shapes depending on the number of nodes subdividing the length of the member. In the simplest cases the configuration of the vibrating system can be determined by one quantity only. Such systems are called systems having one degree of freedom. Let us consider the case shown in Fig. 1. If the arrangement be such that only vertical displacements of the weight W are possible and the mass of the spring be small in compari- son with that of the weight W, the system can be considered as having one degree of freedom. The configuration will be determined completely by the vertical displacement of the weight. By an impulse or a sudden application and removal of an external force vibrations of the system can be produced. Such vibrations which are maintained by the elastic force in the spring alone are called free or natural vibrations. An analytical expression for these FIG. 1. vibrations can be found from the differential equation of motion, which always can be written down if the forces acting on the moving body are known. Let k denote the load necessary to produce a unit extension of the spring. This quantity is called spring constant. If the load is measured in pounds and extension in inches the spring constant will be obtained in Ibs. per in. The static deflection of the spring under the action of the weight W will be </li> <li> 2 VIBRATION PROBLEMS IN ENGINEERING Denoting a vertical displacement of the vibrating weight from its position of equilibrium by x and considering this displacement as positive if it is in a downward direction, the expression for the tensile force in the spring cor- responding to any position of the weight becomes F = W + kx. (a) In deriving the differential equation of motion we will use Newton's prin- ciple stating that the product of the mass of a particle and its acceleration is equal to the force acting in the direction of acceleration. In our case the mass of the vibrating body is W/g, where g is the acceleration due to gravity; the acceleration of the body W is given by the second derivative of the displacement x with respect to time and will be denoted by x] the forces acting on the vibrating body are the gravity force W, acting down- wards, and the force F of the spring (Eq. a) which, for the position of the weight indicated in Fig. 1, is acting upwards. Thus the differential equa- tion of motion in the case under consideration is x = W-(W + kx). (1) a This equation holds for any position of the body W. If, for instance, the body in its vibrating motion takes a position above the position of equilib- rium and such that a compressivc force in the spring is produced the expres- sion (a) becomes negative, and both terms on the right side of eq. (1) have the same sign. Thus in this case the force in the spring is added to the gravity force as it should be. Introducing notation tf = ^ = 4- MP W .,' (2) differential equation (1) can be represented in the following form x + p 2 x = 0. (3) This equation will be satisfied if we put x = C cos pt or x = 2 sin pt, where C and 2 are arbitrary constants. By adding these solutions the general solution of equation (3) will be obtained: x = Ci cos pt + 2 sin pt. (4) It is seen that the vertical motion of the weight W has a vibratory charac- </li> <li> HARMONIC VIBRATIONS 3 ter, since cos...</li></ul>

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