Search -
Theory of the Vibration of Rotating Shafts
Theory of the Vibration of Rotating Shafts Author:Warren Weaver Purchase of this book includes free trial access to www.million-books.com where you can read more than a million books for free. This is an OCR edition with typos. Excerpt from book: CHAPTER 5 45 THE METHOD OF BEHRENS 1 STATEMENT OF THE PROBLEM AND PRELIMINARY DISCUSSION Introductory note. This chapter consists of a study of tne anal... more »ysis presented by Behrens in a 3ottingen disertation published in "Zeitschrift fur xathematik und Physik", Vol. 5?, 1?11.- The results given were obtained by him in that article, out in many instances the details of analysis will oe cnanged in the hope of gaining clearness. The content of Art. 8) is not to be found in Behren's paper, but was added to make more definite the general methods being employed, and to introduce the notation. The only other matter of any special importance that is given here that is not contained in Behrens1 article is the proof of tne tneorem ( 2. Art.? ) concerning change of var - iables in canonical equations. The proof of this theorem could not oe found in any of the standard texts to which we have had access, and while a statement of it was found in Jacobi's original works there is no proof even in that source, so that it seems worth while to give it in full. 1) A slender shar't is thougnt of as being in an initial state of rotation. Whatever forces may have been necessary to produce this rotation are no longer acting. 3ravity and all losses, ooth of external and internal friction, will be neglected. At the middle point of tnis shaft is rigidly attached to it a very thin disk, the plane of which is perpendicular to the center line of the shaft. The disk is eccentrically attached to the shaft: i.e. its center of gravity does not coincide with the point at which the center line of the shaft pierces the disk. Given the initial oosition and velocity of the center of gravity our problem consists of studying the motion of that point. 2) The equations of motion of the center of gravity will no...« less