Rotordynamics

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 It has been suggested that Rotor dynamics be merged into this article or section. (Discuss)

Rotordynamics is a specialized branch of applied mechanics concerned with the behavior and diagnosis 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 and influenced by internal phenomena 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 a maximum that is called a critical speed. This amplitude is commonly excited by unbalance of the rotating structure; everyday examples include engine balance and tire balance. If the amplitude of vibration at these critical speeds is excessive catastrophic failure[1] occurs. In addition to this, turbomachinery often develop instabilities which are related to the internal makeup of turbomachinery, and which must be corrected. This is the chief concern of engineers who design large rotors.

Contents

[edit] Basic Principles

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

\begin{matrix} M\ddot{q}(t)+(C+G)\dot{q}(t)+(K){q}(t)&=&f(t)\\ \end{matrix}

where:

M is the symmetric mass matrix

C is the symmetric damping matrix

G is the skew-symmetric gyroscopic matrix

K is the symmetric bearing or seal stiffness matrix

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

The gyroscopic matrix G is proportional to spin speed Ω. The general solution to the above equation involves complex eigenvectors which are spin speed dependent. Engineering specialists in this field rely on the Campbell Diagram to explore these solutions.

An interesting feature of the rotordynamic system of equations are the off-diagonal terms of stiffness, damping, and mass. These terms are called cross-coupled stiffness, cross-coupled damping, and cross-coupled mass. When there is a positive cross-coupled stiffness, a deflection will cause a reaction force opposite the direction of deflection to react the load, and also a reaction force in the direction of positive whirl. If this force is large enough compared with the available direct damping and stiffness, the rotor will be unstable. When a rotor is unstable it will typically require immediate shutdown of the machine to avoid catastrophic failure.

[edit] Campbell Diagram

Campbell Diagram for a Simple Rotor
Campbell Diagram for a Simple Rotor

The Campbell diagram of a simple rotor system is shown on the right. The red and blue curves show the backward whirl (BW) and forward whirl (FW) modes, which diverge as the spin speed increases. When the BW frequency or the FW frequency equal the spin speed Ω, indicated by the intersections A and B with the synchronous spin speed line, the response of the rotor may show a peak. This is called a critical speed.

[edit] Jeffcott Rotor

The Jeffcott rotor (also known as the DeLaval rotor in Europe) is a simplified lumped parameter model used to solve these equations. Jeffcott's rotor was a mathematical tool that did not reflect actual rotor mechanics.

[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. August 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. The most prevalent method used today for rotordynamics analysis is the Finite Element Method.

Modern computer models have been commented on in a quote attributed to Dara Childs, "the quality of predictions from a computer code has more to do with the soundness of the basic model and the physical insight of the [analyst]. ... Superior algorithms or computer codes will not cure bad models or a lack of engineering judgment."

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

[edit] Software

There are many software packages that are capable of solving the rotordynamic system of equations. Rotordynamic specific codes are more versatile for design purposes. These codes make it easy to add bearing coefficients, side loads, and many other items only a rotordynamicist would need. The non-rotordynamic specific codes are full featured FEA solvers, and have many years of development in their solving techniques. The non-rotordynamic specific codes can also be used to calibrate a code designed for rotordynamics.

Rotordynamic Specific codes:

Non-rotordynamic specific codes:

  • Ansys - Version 11 workbench and classic is capable of solving the rotordynamic equations (3-D/2-D and beam element)
  • Nastran - Finite element based (3-D/2-D and beam element)

[edit] See also

[edit] References

  • Chen, W. J., Gunter, E. J. (2005). Introduction to Dynamics of Rotor-Bearing Systems. ISBN 1-4120-5190-8.  uses DyRoBeS
  • Childs, D. (1993). Turbomachinery Rotordynamics Phenomena, Modeling, & Analysis. Wiley. ISBN 0-471-53840-X. 
  • Genta, G. (2005). Dynamics of Rotating Systems. Springer. ISBN 978-0-387-20936-4. 
  • 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. 
  • Krämer, E. (1993). Dynamics of Rotors and Foundations. Springer-Verlag. ISBN 0-387-55725-3. 
  • Lalanne, M.,Ferraris, G. (1998). Rotordynamics Prediction in Engineering, Second Edition. Wiley. ISBN 978-0-471-97288-4. 
  • Muszyńska, Agnieszka (2005). Rotordynamics. CRC Press. ISBN 978-0-8247-2399-6. 
  • Nelson, F. (June, 2003). "A Brief History of Early Rotor Dynamics". Sound and Vibration. 
  • Nelson, F. (July 2007). "Rotordynamics without Equations". International Journal of COMADEM 10. ISSN 1363-7681. 
  • Yamamoto, T.,Ishida, Y. (2001). Linear and Nonlinear Rotordynamics. Wiley. ISBN 978-0-471-18175-0. 
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