Torsional Vibration of Turbomachinery
Including application case studies and in-depth interpretation of computer-generated results, this book offers practical answers to preventing excessive damage, failure and noise due to vibrations in machines.
TABLE OF CONTENTS Torsional Vibration of Turbomachinery
Appendix B: Mathematical Derivation of Torsion Parameters
This appendix derives some of the fundamental torsional parameters used in the text, including the derivations of the results shown in Table 6.1.
Polar Moment of Inertia
The polar moment of inertia of a point mass m located a distance r from the rotational axis is expressed by mr 2.
Figure B.1 shows the cross section of a uniform shaft with an annular cross section. The internal and external radii of the shaft are R i and R o , respectively. The radius of an elemental annular ring is r and the ring s elemental thickness is ? r.
Figure B.1: Shaft cross section.
If the shaft has a length of L and a density ?, then the mass of the annular element is 2 ? r ?r Lp. The shaft s polar moment of inertia is by definition the product of the elemental mass times the square of its distance to the polar (rotational) axis r. Thus the shaft s polar moment of inertia = 2 ?r 3 ?r L ? .
The total polar moment of inertia of the uniform shaft segment is obtained by integration of the elemental value from the inner radius to the outer radius of the shaft, giving
Therefore 
Hence 
where D i and D o are the inner and outer diameters, respectively.
Torsional Stiffness and Stress
Figure B.2 shows the outside surface of a short segment of cylindrical shaft...
Copyright The McGraw-Hill Companies, Inc. 2004 under license agreement with Books24x7
