Vibration problems in engineering by s timoshenko

VIBRATION PROBLEMS IN ENGINEERING BY S TIMOSHENKO Professor of Theoretical and Engineering Mechanics Stanford University SECOND EDITIONFIFTH PRINTING D NEW YORK VAN NOSTRAND COMPANY, 250 FOURTH AVENUE INC 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' J, Auyu^t, t'^44, t PRINTED IN THE USA A/ 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 With this in view, the purposes written and considerably enlarged first part of the book was entirely re- 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: article on the method of successive approximations is added and it shown how the method can be used in discussing free and forced vibraThe third chapter is tions of systems with non-linear characteristics made more complete by including in it a general discussion of the equation of vibratory motion of systems with variable spring characteristics The tems an is 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 PREFACE TO THE FIRST EDITION With the increase of size analysis of vibration problems and velocity in modern machines, the 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 In experience with vibration of machines and structures in service has the author followed on this the lectures vibration book, developing 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 The general theory of free and of systems with one degree of freedom 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 complicated 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 systems 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 investigation of vibrations in electric locomotives with side-rod drive * Theory of Elasticity, Vol II (1916) St Petersburg, Russia v PREFACE TO THE FIRST EDITION vi In chapter three, systems with several degrees of freedom are conThe 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 sidered 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 shafts : 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 instruments 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 machines which were investigated by the company's He takes this opportunity to thank, also, the numerous engineers of vibration in who have him in various ways in the preparation of the J M Lessells, J Ormondroyd, and J P Messr manuscript, particularly Den Hartog, who have read over the complete manuscript and have made friends assisted many valuable suggestions He is indebted, also, to Mr F C Wilharm for the preparation of drawings, and to the Van Nostrand Company for their care in the publication oi the book S ANN ARBOR, MICHIGAN, May 22, 1928 TIMOSHENKO CONTENTS CHAPTER I PAGE HARMONIC VIBRATIONS OF SYSTEMS HA VINO ONE DEGREE OF FREEDOM Free Harmonic Vibrations Torsional Vibration Forced Vibrations Instruments for Investigating Vibrations Spring Mounting of Machines Other Technical Applications 19 24 /V Damping ^78 Free Vibration with Viscous Damping Forced Vibrations with Viscous Damping 10 Spring Mounting of Machines with Damping Considered 11 Free Vibrations with Coulomb's Damping 12 Forced Vibrations with Coulomb's Damping and other Kinds of Damping 13 Machines for Balancing 14 General Case of Disturbing Force v/15 Application of Equation of Energy in Vibration Problems 18 Rayleigh Method Critical Speed of a Rotating Shaft General Case of Disturbing Force 19 Effect of 16 17 Low Spots on Deflection of Rails 51 54 57 62 64 74 83 92 98 107 110 20 Self-Excited Vibration CHAPTER 26 30 32 38 IT VIBRATION OF SYSTEMS WITH NON-LINEAR CHARACTERISTICS 21 Examples of Non-Linear Systems Systems with Non-linear Restoring Force 22 Vibrations of 114 119 23 Graphical Solution 121 24 Numerical Solution 126 25 Method of Successive Approximations Applied to Free Vibrations Forced Non-Linear Vibrations 137 26 CHAPTER 129 111 SYSTEMS WITH VARIABLE SPRING CHARACTERISTICS 27 151 Examples of Variable Spring Characteristics of the Equation of Vibratory Motion with Variable Spring 28 Discussion 160 Characteristics 29 Vibrations in the Side Rod Drive System vii of Electric Locomotives 167 CONTENTS via CHAPTER IV PAGE SYSTEMS HAVING SEVERAL DEGREES OF FREEDOM 30 d'Alembert's Principle and the Principle of Virtual Displacements 31 Generalized Coordinates and Generalized Forces 182 185 Equations 189 Spherical Pendulum 34 Free Vibrations General Discussion 192 35 Particular Cases 206 32 Lagrange's 33 36 The 194 208 213 216 222 Forced Vibrations 37 Vibration with Viscous 38 Stability of Damping Motion Caused by Hysteresis 39 Whirling of a Rotating Shaft 229 240 40 Vibration of Vehicles 41 Damping Vibration Absorber CHAPTER V TORSIONAL AND LATERAL VIBRATION OF SHAFTS 42 Free Torsional Vibrations of Shafts 43 Approximate Methods of Calculating Frequencies of Natural Vibrations Forced Torsional Vibration of a Shaft with Several Discs 44 45 Torsional Vibration of Diesel 46 Damper with Engine Crankshafts Solid Friction 47 Lateral Vibrations of Shafts on Many Supports 48 Gyroscopic Effects on the Critical Speeds of Rotating Shafts 49 Effect of Weight of Shaft and Discs on the Critical Speed 50 Effect of Flexibility of Shafts on the Balancing of Machines CHAPTER 253 258 265 270 274 277 290 299 303 VI VIBRATION OF ELASTIC BODIES 51 Longitudinal Vibrations of Prismatical 52 Vibration of a Bars 307 317 325 Bar with a Load at the End 53 Torsional Vibration of Circular Shafts 54 Lateral Vibration of Prismatical 55 The V56 Free Vibration of Jbl Other >/58 Bars 331 and Rotatory Inertia a Bar with Hinged Ends Effect of Shearing Force End Fastenings of a Forced Vibration Beam with Supported Ends 59 Vibration of Bridges 60 Effect of Axial Forces 61 Vibration of 62 Ritz on Lateral Vibrations Elastic Foundation Beams on Method 63 Vibration of Bars of Variable Cross Section 64 Vibration of Turbine Blades 337 338 342 348 358 364 368 370 376 382 CONTENTS ix PAGE 65 Vibration of Hulls of Ships Impact of Bars 67 Longitudinal Impact of Prismatical Bars *8 Vibration of a Circular Ring 06 Lateral '69 Vibration of Membranes 411 421 70 Vibration of Plates 71 Vibration of 388 392 397 405 Turbine Discs 435 APPENDIX VIBRATION MEASURING INSTRUMENTS General Frequency Measuring Instruments The Measurement of Amplitudes Seismic Vibrographs Torsiograph Torsion Meters Strain Recorders AUTHOR INDEX SUBJECT INDEX 443 443 444 448 452 453 457 463 467 CHAPTER I HARMONIC VIBRATIONS OF SYSTEMS HAVING ONE DEGREE OF FREEDOM 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 position 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 In the simplest cases the configuration of the vibrating of the member can be determined by one quantity only Such systems are called system systems having one degree of freedom Let us consider the case shown in Fig If the arrangement be such are that only vertical displacements of the weight W possible and the mass be small in comparison with that of the weight W, the system can be The considered as having one degree of freedom 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 of the spring the elastic force in the spring alone are called free or FIG natural vibrations An analytical expression for these 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 will be W VIBRATION PROBLEMS IN ENGINEERING Denoting a vertical displacement of the vibrating weight from its position by x and considering this displacement as positive if it is in a downward direction, the expression for the tensile force in the spring corresponding to any position of the weight becomes of equilibrium 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 In our case the is equal to the force acting in the direction of acceleration W mass of the vibrating body is /g, where g is the acceleration due to is given by the second derivative gravity; the acceleration of the body of the displacement x with respect to time and will be denoted by x] the W on the vibrating body are the gravity force W, acting downwards, 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- forces acting tion of motion in the case under consideration = x is 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 equilibrium and such that a compressivc force in the spring is produced the expression (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 P ^ = 4W M (2) ,' differential equation (1) can be represented in the following form x This equation where C\ and will be satisfied + if p x = we put x (3) = C\ cos pt or x = sin pt, By adding these solutions the solution of be will obtained: general equation (3) are arbitrary constants x It is seen that the vertical = Ci cos pt motion + sin pt of the weight (4) W has a vibratory charac- SUBJECT INDEX Bars Continued Forced Vibrations (supported ends) moving constant force, 352 moving pulsating force, 356 pulsating force, 349 Beats, 18, 236 Accelerometer, 22, 462 d'Alembert's Principle, 182 Amplitude, definition of, frequency diagram, 41 measurement of, 76, 444 Bridges, Vibration of Automobile Vibration, 229 Axial Forces, effect on lat 364 impact of unbalanced weights, 360 irregularities of track; flats, etc., 364 moving mass, 358 vibr of bars, B Balancing, 62 effect of shaft Karelitz Cantilever Beam, Vibration, 86, 344 flexibility, method of, 303 Centrifugal Balancing Machines Akimoff's, 68 Conical Crank 344 clamped ends, 343 cantilever, 86, differential equation, effect of axial forces, Rod Vibration, 380 Constraint, Equations of, 183 Crank Drive, Inertia of, 272 Lateral Vibrations of Shaft, Torsional Flexibility, 270 Critical 332 364 effect of shear, etc., 337, 6, 85, 338, Damping, 37 Critical Regions, 175 Critical Speeds of Shafts analytical determination of, 277 effect of gravity, 299 341 342 of three bearing set, 285 graphical determination of, 95, 283 348 example many supports, 345 one end clamped, one supported, 345 on elastic foundation, 368 variable section, cantilever, 378 variable section, free ends, 381 with variable on frequency, Coefficient of Friction, 31 Collins Micro Indicator, 29 Ears hinged ends, effect Circular Frequency, Lawaczek-Heymann's, 64 Soderberg's, 69 free ends, Force, 366 71 flexibility, 153, gyroscopic effect, 290 rotating shaft, several discs, 94 rotating shaft, single disc, 92 variable flexibility, 153 Critical Speed of Automobile, 237 376 Longitudinal Vibrations of cantilever with loaded end, 317 differential equation, 309 force suddenly applied, 323 struck at the end, 397 Damper with Damping Solid Friction, 274 constant damping, 30 energy absorption due trigonometric series solution, 309 467 to, 45 SUBJECT INDEX 468 Continued Damping proportional to velocity, 32 in torsional vibration, 271 Degree of Geared Systems, torsional vibration, 256 definition, Freedom, Diesel Engine, Torsional vibration, 270 Discs, Turbine, 435 Generalized Coordinate, 185 Generalized Force, 187 Governor, Vibration, 219 Graphical Integration, 121 Grooved Rotor, 98 Gyroscopic Effects, 290 Disturbing Force, 14 general case of, 98 Dynamic Vibration Absorber, 240 Harmonic Motion, definition, Hysteresis loop, 32, 223 Elastic Foundation, Vibration of bars on, 368 Energy Impact absorbed by damping, 49 method of calculation, 74 Equivalent Disc, 273 on bridges, 358 on bars, 392 longitudinal on bars, 397 Indicator, steam engines, 28 Inertia of Crank Drive, 272 Equivalent Shaft Length, 10, effect lateral 271 Flexible Bearings with rigid rotor, 296 Forced Vibrations definition, 15 Lagrange's equations, 189 Lateral Vibration of bars, 332 Lissajous Figures, 447 Logarithmical decrement, 35 Longitudinal Vibration of bars, 307 general theory, 208 non-linear, 137 torsional, 265 with damping, 37, 57 Foundation Vibration, 24, 51, 101 Frahm Tachometer, 27 Frame, Vibration of circular, 405, 410 rectangular, 90 Free Vibrations definition, M Magnification Factor, 15, 40, 59 Membranes circular, rectangular, 412 Modes general theory, 194 418 general, 411 of Vibration, 197 principal, 198 with Coulomb damping, 54 with viscous damping, 32 N Frequency circular, definition, Natural Vibrations, equation, 197, 198 measurement of, 448 Fullarton Vibrometer, 28, 443 Fundamental Type of Vibration, Nodal Section, 11 Non-Linear Restoring Force, 119 Non-Linear Systems, 114 200 Normal Coordinates, 124, 127 Normal Functions, 309 SUBJECT INDEX Oscillator, 469 Seismic Instruments, 19 Self-Excited Vibrations, 110 26 Ships, Hull Vibration of, 388 Side Pallograph, 80 Rod Drive, 167 Spring Characteristic Variable, 151 Spring Constant, Spring Mounting, 24, 51 Pendulum double, 203 spherical, 192 Stability of Motion, 216 variable length, 155 Strain Recorder Cambridge, 457 magnetic 458 telemeter, 459 Period, definition, Phase definition, 6, 16 diagram, 42, 61 with damped vibration 42, 60 Phasometer, 74 Sub-Harmonic Resonance, 149 Plates 426 clamped at boundary, 428 effect of stretch of middle surface, 431 circular, free, 424 general, 421 rectangular, 422 Principal Coordinates, 197 Tachometer, Frahm, 27 Telemeter, 459 Torsiograph, 452 Torsion Meter Amsler, 454 Moullin, 453 Vieweg, 456 Torsional Vibrations effect of many mass discs, of shaft, 325 255 single disc, Rail Deflection, 107 Rail Vibration, 256 two discs, 11 three discs, 254 Rayleigh Method, 83 Transient, 49 in torsional vibration, 260 Transmissibility, 52 ~^ Regions of Critical Speed, 175 Resonance, definition, 15 Transmission lines vibration, 112 Turbine Blades, 382 Ring Complete Turbine Discs, 435 flexural vibration, 408 radial vibration, 405 torsional vibration, 407 Incomplete, 410 Ritz Method, 370, 424 Unbalance, definitions dynamic, 63 static, 63 Universal Recorder, 335 Shafts critical speed of, 282 277 torsional vibrations, 253 lateral vibrations, Schlingertank, 252 Variable Cross Section cantilever, 378 free ends, 381 Variable Flexibility, 151 470 SUBJECT INDEX Vehicle Vibration, 229 Vibration Absorber, 240 Virtual Displacement, Principle Viscosity, 31 Vibrograph Cambridge, 448 Geiger, 449 Viscous Damping, 32, 213 W theory, 19 Vibrometer Fullarton, 443 Wedge, 378 Vibration Specialty, 445 Whirling of Shafts, 222 of, 182 ... its application in the solution of such engineering problems as: the vibration of vehicles, the torsional vibration of shafts, whirling speeds sidered on several supports, and vibration absorbers... HARMONIC VIBRATIONS OF SYSTEMS HA VINO ONE DEGREE OF FREEDOM Free Harmonic Vibrations Torsional Vibration Forced Vibrations Instruments for Investigating Vibrations Spring Mounting of Machines Other... Turbine Discs 435 APPENDIX VIBRATION MEASURING INSTRUMENTS General Frequency Measuring Instruments The Measurement of Amplitudes Seismic Vibrographs Torsiograph Torsion Meters Strain Recorders