VIBRATION PROBLEMS IN ENGINEERING by S. TIMOSHENKO Professor of Theoretical and Engineering Mechanics Stanford University. Second edition published in may 1937. PREFACE TO THE SECOND EDITION: In the preparation of the manuscript for the second edition of the book, the authors 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 vibrations 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 problems of the first chapter. 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...