Rotordynamics

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Rotordynamics is a specialized branch of applied mechanics concerned with the behavior of rotating structures. It is commonly used to analyze the behavior of structures ranging from jet engines and steam turbines to auto engines and computer disk storage. At its most basic level rotordynamics is concerned with one or more mechanical structures (rotors) supported by bearings that rotate around a single axis. The supporting structure is called a stator. As the speed of rotation increases the amplitude of vibration often passes through maxima that are called critical speeds. If the amplitude of vibration at these critical speeds are excessive catastrophic failure[1] occurs. This is the chief concern of engineers who design large rotors.

[edit] Basic Principles

The equation of motion, in generalized matrix form, for an axially symmetric rotor rotating at a constant spin speed Ω is

where:

M is the symmetric mass matrix

C is the symmetric damping matrix

G is the skew-symmetric gyroscopic matrix

K is the symmetric stiffness matrix

H is the skew-symmetric circulatory matrix

in which q is the generalized coordinates of the rotor in inertial coordinates and f is a forcing function.

Both the gyroscopic matrix G and the circulatory matrix H are proportional to spin speed Ω.

The general solution to the above equation involves complex eigenvectors which are spin speed dependent. The Jeffcott rotor is a simplified lumped parameter model used to solve these equations. Engineering specialists in this field rely on the Campbell Diagram to explore these solutions.

[edit] History

The history of rotordynamics is replete with the interplay of theory and practice. W. J. M. Rankine first performed an analysis of a spinning shaft in 1869, but his model was not adequate and he predicted that supercritical speeds could not be attained. In 1895 Dunkerley published an experimental paper describing supercritical speeds. Carl Gustaf De Laval, a Swedish engineer, ran a steam turbine to supercritical speeds in 1889, and Kerr published a paper showing experimental evidence of a second critical speed in 1916.

Henry Jeffcott was commissioned by the Royal Society of London to resolve the conflict between theory and practice. He published a paper now considered classic in the Philosophical Magazine in 1919 in which he confirmed the existence of stable supercritical speeds. Föppl published much the same conclusions in 1895, but history largely ignored his work.

Between the work of Jeffcott and the start of World War II there was much work in the area of instabilities and modeling techniques culminating in the work of Prohl and Myklestad which led to the Transfer Matrix Method (TMM) for analyzing rotors.

Prof. F. Nelson has written extensively on the history of rotordynamics and most of this section is based on his work.

[edit] References

• Genta, G. (2005). Dynamics of Rotating Systems. Springer. ISBN 0-387-20936-0.
• Jeffcott, H. H. (1919). "The Lateral Vibration Loaded Shafts in the Neighborhood of a Whirling Speed. - The Effect of Want of Balance". Philosophical Magazine 37.
• Lalanne, M.,Ferraris, G. (1998). Rotordynamics Prediction in Engineering, Second Edition. Wiley. ISBN 0-471-97288-6.
• Muszyńska, A. (2005). Rotordynamics. CRC Press. ISBN 0-8247-2399-6.
• Nelson, F. (June, 2003). "A Brief History of Early Rotor Dynamics". Sound and Vibration.
• Yamamoto, T.,Ishida, Y. (2001). Linear and Nonlinear Rotordynamics. Wiley. ISBN 0-471-18175-7.