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| '''Rotor dynamics''' 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 turbine]]s to auto engines and computer [[disk storage]]. At its most basic level rotor dynamics is concerned with one or more mechanical structures ([[Rotor (electric)|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 then catastrophic failure occurs. In addition to this, turbo machinery often develop instabilities which are related to the internal makeup of turbo machinery, and which must be corrected. This is the chief concern of engineers who design large rotors.
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| Rotating machinery produces vibrations depending upon the structure of the mechanism involved in the process. Any faults in the machine can increase or excite the [[vibration]] signatures. [[Vibration]] behavior of the machine due to imbalance is one of the main aspects of rotating machinery which must be studied in detail and considered while designing. All objects including rotating machinery exhibit natural frequency depending on the structure of the object. The critical speed of a rotating machine occurs when the rotational speed matches its natural frequency. The lowest speed at which the natural frequency is first encountered is called the first [[critical speed]], but as the speed increases additional critical speeds are seen. Hence, minimizing rotational unbalance and unnecessary external forces are very important to reducing the overall forces which initiate [[Resonance energy transfer|resonance]]. When the vibration is in [[Resonance energy transfer|resonance]] it creates a destructive energy which should be the main concern when designing a rotating machine. The objective here should be to avoid operations that are close to the critical and pass safely through them when in acceleration or deceleration. If this aspect is ignored it might result in loss of the equipment, excessive wear and tear on the machinery, catastrophic breakage beyond repair or even human injury and loss of lives.
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| The real dynamics of the machine is difficult to model theoretically. The calculations are based on simplified models which resemble various structural components ([[lumped parameters]] models), equations obtained from solving models numerically ([[Rayleigh-Ritz method]]) and finally from the [[Finite Element Method]] (FEM), which is another approach for modelling and analysis of the machine for natural frequencies. On any machine prototype it is tested to confirm the precise frequencies of resonance and then redesigned to assure that resonance does not occur.
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| ==Basic principles==
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| The [[equation of motion]], in generalized [[matrix(mathematics)|matrix]] form, for an axially symmetric rotor rotating at a constant spin speed Ω is
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| :<math>
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| \begin{matrix}
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| \bold {M}\ddot{\bold{q}}(t)+(\bold{C}+\bold{G})\dot{\bold{q}}(t)+(\bold{K}+\bold{N}){\bold{q}}(t)&=&\bold{f}(t)\\
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| \end{matrix}
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| </math>
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| where:
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| '''M''' is the [[symmetric matrix|symmetric]] [[Mass matrix]]
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| '''C''' is the [[symmetric matrix|symmetric]] [[damping matrix]]
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| '''G''' is the [[skew-symmetric matrix|skew-symmetric]] gyroscopic matrix
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| '''K''' is the [[symmetric matrix|symmetric]] bearing or seal stiffness matrix
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| '''N''' is the gyroscopic matrix of deflection for inclusion of e.g., centrifugal elements.
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| in which '''q''' is the generalized coordinates of the rotor in inertial coordinates and '''f''' is a forcing function, usually including the unbalance.
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| The gyroscopic matrix '''G''' is proportional to spin speed Ω.
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| The general solution to the above equation involves [[complex number|complex]] [[eigenvectors]] which are spin speed dependent.
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| Engineering specialists in this field rely on the [[Campbell Diagram]] to explore these solutions.
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| 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.
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| ==Campbell diagram==
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| [[File:Campbelldiagram.png|thumb|300px|[[Campbell Diagram]] for a Simple Rotor]]
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| The [[Campbell Diagram|Campbell diagram]], also known as "Whirl Speed Map" or a "Frequency Interference
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| Diagram", of a simple rotor system is shown on the right. The pink and blue curves show the backward whirl (BW) and forward whirl (FW) modes, respectively, 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]].
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| ==Jeffcott rotor==
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| The Jeffcott rotor (named after Henry Homan Jeffcott), also known as the [[Gustaf de Laval|de Laval]] rotor in Europe, is a simplified lumped parameter model used to solve these equations. The Jeffcott rotor is a mathematical [[idealization]] that may not reflect actual rotor mechanics.
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| ==History==
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| The history of rotordynamics is replete with the interplay of theory and practice. [[William John Macquorn Rankine|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. [[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.
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| 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.
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| 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 [[Nils Otto 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]].
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| 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 [[systems analyst|analyst]]. ... Superior algorithms or computer codes will not cure bad models or a lack of engineering judgment."
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| Prof. [[Frederick Nelson|F. Nelson]] has written extensively on the history of rotordynamics and most of this section is based on his work.
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| ==Software==
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| There are many software packages that are capable of solving the rotor dynamic system of equations. Rotor dynamic 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-rotor dynamic specific codes are full featured FEA solvers, and have many years of development in their solving techniques. The non-rotor dynamic specific codes can also be used to calibrate a code designed for rotor dynamics.
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| Rotordynamic specific codes:
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| * SAMCEF ROTOR,([http://www.lmsintl.com/rotor-dynamics SAMCEF]) - Software Platform for Rotors Simulation (LMS Samtech,A Siemens Business)
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| * MADYN 2000 ([http://www.delta-js.ch/english/software/madyn-2000-for-rotordynamics/rotordynamic-software-madyn-2000.html DELTA JS Inc.]) - Commercial combined finite element (3D Timoshenko beam) lateral, torsional, axial and coupled solver for multiple rotors and gears, various bearings (fluid film, spring damper, magnetic)
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| * iSTRDYN ([http://www.istrdyn.com DynaTech Software LLC]) - Commercial 2-D Axis-symmetric finite element solver
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| * FEMRDYN ([http://dynatechengr.com/femrdyn.htm DynaTech Engineering, Inc.]) - Commercial 1-D Axis-symmetric finite element solver
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| * Dyrobes ([http://www.dyrobes.com Eigen Technologies, Inc.]) - Commercial 1-D beam element solver
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| * RIMAP ([http://ritec-eg.com/Products/RITEC/RIMAP_Rotordynamic_Analysis_Software_Package.html RITEC]) - Commercial 1-D beam element solver
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| * XLRotor ([http://www.xlrotor.com Rotating Machinery Analysis, Inc.]) - Commercial 1-D beam element solver. It provides powerful, fast and accurate tool to perform rotor dynamic modeling and analysis.
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| * ARMD ([http://www.rbts.com Rotor Bearing Technology & Software, Inc.]) - Commercial 1-D beam element solver
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| * XLTRC2 ([http://turbolab.tamu.edu Texas A&M]) - Academic 1-D beam element solver
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| * ComboRotor ([http://www.virginia.edu/romac/ University of Virginia]) - Combined finite element lateral, torsional, axial solver for multiple rotors evaluating critical speeds, stability and unbalance response extensively verified by industrial use
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| * Dynamics R4 ([http://www.alfatran.com Alfa-Tranzit Co. Ltd]) - Commercial software developed for design and analysis of spatial systems
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| * MESWIR ([http://www.imp.gda.pl/index_a.html Institute of Fluid-Flow Machinery, Polish Academy of Sciences]) - Academic computer code package for analysis of rotor-bearing systems whithin the linear and non-linear range
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| * RoDAP ([http://www.dnmco.com D&M Technology]) - Commercial lateral, torsional, axial and coupled solver for multiple rotors, gears and flexible disks(HDD)
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| * ROTORINSA ([http://rotorinsa.insa-lyon.fr/rotorinsa/expertise.php?rub=02, ROTORINSA]) - Commercial finite element software developed by a French engineering school (INSA-Lyon) for analysis of steady-state dynamic behavior of rotors in bending.
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| Non-rotordynamic specific codes:
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| * [[Ansys]] - Version 11 workbench and classic is capable of solving the rotordynamic equations (3-D/2-D and beam element)
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| * [[Nastran]] - Finite element based (3-D/2-D and beam element)
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| * [[SAMCEF]] - Finite element based (3-D/2-D and beam element)
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| ==See also==
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| * [[Axle]]
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| * [[Balancing machine]]
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| * [[Bearing (mechanical)]]
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| * [[Driveshaft]]
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| * [[Exoskeletal engine]]
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| * [[Magnetic bearing]]
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| * [[Turbine]]
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| * [http://spectraquest.com/technote_display/?technote_id=29# Rotordynamic Analysis using XLRotor]
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| ==References==
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| <references/> | |
| *{{cite book | author=Chen, W. J., Gunter, E. J.| title= Introduction to Dynamics of Rotor-Bearing Systems| year=2005 | isbn=1-4120-5190-8 | publisher=Trafford | location=Victoria, BC}} uses DyRoBeS
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| *{{cite book | author=Childs, D. | title=Turbomachinery Rotordynamics Phenomena, Modeling, & Analysis | publisher=Wiley | year=1993 | isbn=0-471-53840-X}}
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| *{{cite book | author=Fredric F. Ehrich (Editor) | title=Handbook of Rotordynamics | publisher=McGraw-Hill | year=1992 | isbn=978-0-07-019330-7}}
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| *{{cite book | author=Genta, G. | title=Dynamics of Rotating Systems| publisher=Springer | year=2005 | isbn=978-0-387-20936-4}}
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| *{{cite journal | author=Jeffcott, H. H. | title=The Lateral Vibration Loaded Shafts in the Neighborhood of a Whirling Speed. - The Effect of Want of Balance| journal=Philosophical Magazine|volume = 37 | year= 1919|series = 6}}
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| *{{cite book | author=Krämer, E.| title=Dynamics of Rotors and Foundations| publisher=Springer-Verlag| year=1993 | isbn=0-387-55725-3}}
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| *{{cite book | author=Lalanne, M., Ferraris, G.| title=Rotordynamics Prediction in Engineering, Second Edition | publisher=Wiley | year=1998 | isbn=978-0-471-97288-4}}
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| *{{cite book | author=Muszyńska, Agnieszka| title=Rotordynamics | publisher=CRC Press | year=2005| isbn=978-0-8247-2399-6}}
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| *{{cite journal | author=Nelson, F. | title=A Brief History of Early Rotor Dynamics | journal=Sound and Vibration |date=June 2003}}
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| *{{cite journal | author=Nelson, F. | title=Rotordynamics without Equations| journal=International Journal of COMADEM|volume = 10 |date=July 2007|series = 3 | issn=1363-7681}}
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| *{{cite book | author=Nelson, F.| title=An Introduction to Rotordynamics | year=2011 | volume=SVM-19 [http://www.savecenter.org/Products.html]}}
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| *{{cite book | author=Lalanne, M., Ferraris, G.| title=Rotordynamics Prediction in Engineering, Second Edition | publisher=Wiley | year=1998 | isbn=978-0-471-97288-4}}
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| *{{cite book | author=Vance, John M.| title=Rotordynamics of Turbomachinery | publisher=Wiley | year=1988 | isbn=0-471-80258-1}}
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| *{{cite book | author=Yamamoto, T., Ishida, Y. | title=Linear and Nonlinear Rotordynamics | publisher=Wiley | year=2001 | isbn=978-0-471-18175-0}}
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| * Ganeriwala, S., Mohsen N (2008). ''Rotordynamic Analysis using XLRotor.'' [http://spectraquest.com/technote_display/?technote_id=29 SQI03-02800-0811]
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| [[Category:Dynamics]]
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