Rotordynamics, also known as rotor dynamics, is a specialized branch of
applied mechanics
Applied mechanics is the branch of science concerned with the motion of any substance that can be experienced or perceived by humans without the help of instruments. In short, when mechanics concepts surpass being theoretical and are applied and e ...
concerned with the behavior and diagnosis of rotating structures. It is commonly used to analyze the behavior of structures ranging from
jet engines
A jet engine is a type of reaction engine discharging a fast-moving jet of heated gas (usually air) that generates thrust by jet propulsion. While this broad definition can include rocket, water jet, and hybrid propulsion, the term typicall ...
and
steam turbine
A steam turbine is a machine that extracts thermal energy from pressurized steam and uses it to do mechanical work on a rotating output shaft. Its modern manifestation was invented by Charles Parsons in 1884. Fabrication of a modern steam turbin ...
s to auto engines and computer
disk storage
Disk storage (also sometimes called drive storage) is a general category of storage mechanisms where data is recorded by various electronic, magnetic, optical, or mechanical changes to a surface layer of one or more rotating disks. A disk drive is ...
. At its most basic level, rotor dynamics 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
The stator is the stationary part of a rotary system, found in electric generators, electric motors, sirens, mud motors or biological rotors. Energy flows through a stator to or from the rotating component of the system. In an electric mot ...
. As the speed of rotation increases the amplitude of vibration often passes through a maximum that is called a
critical speed In solid mechanics, in the field of rotordynamics, the critical speed is the theoretical angular velocity that excites the natural frequency of a rotating object, such as a shaft, propeller, leadscrew, or gear. As the speed of rotation approache ...
. This amplitude is commonly excited by imbalance of the rotating structure; everyday examples include
engine balance
Engine balance refers to how the forces (resulting from combustion or rotating/reciprocating components) are balanced within an internal combustion engine or steam engine. The most commonly used terms are ''primary balance'' and ''secondary bal ...
and
tire balance
Tire balance, also called tire unbalance or tire imbalance, describes the distribution of mass within an automobile tire or the entire wheel (including the rim) on which it is mounted.
When the wheel rotates, asymmetries in its mass distribution ...
. If the amplitude of vibration at these critical speeds is excessive, then
catastrophic failure
A catastrophic failure is a sudden and total failure from which recovery is impossible. Catastrophic failures often lead to cascading systems failure. The term is most commonly used for structural failures, but has often been extended to many oth ...
occurs. In addition to this,
turbomachinery
Turbomachinery, in mechanical engineering, describes machines that transfer energy between a rotor and a fluid, including both turbines and compressors. While a turbine transfers energy from a fluid to a rotor, a compressor transfers energy from ...
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.
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
Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force.
The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all pa ...
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 which are the multiples of the natural frequency. Hence, minimizing rotational unbalance and unnecessary external forces are very important to reducing the overall forces which initiate
resonance
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...
. When the vibration is in 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.
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
The lumped-element model (also called lumped-parameter model, or lumped-component model) simplifies the description of the behaviour of spatially distributed physical systems, such as electrical circuits, into a Topology (electrical circuits), t ...
models), equations obtained from solving models numerically (
Rayleigh–Ritz method The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz.
The name Rayleigh–Ritz is being debated ...
) and finally from the
finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
(FEM), which is another approach for modelling and analysis of the machine for natural frequencies. There are also some analytical methods, such as the distributed transfer function method, which can generate analytical and closed-form natural frequencies, critical speeds and unbalanced mass response. On any machine prototype it is tested to confirm the precise frequencies of resonance and then redesigned to assure that resonance does not occur.
Basic principles
The
equation of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
, in generalized
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
form, for an axially symmetric rotor rotating at a constant spin speed Ω is
:
where:
:M is the
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
Mass matrix
In analytical mechanics, the mass matrix is a symmetric matrix that expresses the connection between the time derivative \mathbf\dot q of the generalized coordinate vector of a system and the kinetic energy of that system, by the equation
:T ...
:C is the symmetric
damping matrix
:G is the
skew-symmetric gyroscopic matrix
:K is the symmetric bearing or seal stiffness matrix
:N is the gyroscopic matrix of deflection for inclusion of e.g., centrifugal elements.
in which q is the generalized coordinates of the rotor in inertial coordinates and f is a forcing function, usually including the unbalance.
The gyroscopic matrix G is proportional to spin speed Ω.
The general solution to the above equation involves
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
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.
Campbell diagram
The
Campbell diagram, also known as "Whirl Speed Map" or a "Frequency Interference
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 In solid mechanics, in the field of rotordynamics, the critical speed is the theoretical angular velocity that excites the natural frequency of a rotating object, such as a shaft, propeller, leadscrew, or gear. As the speed of rotation approache ...
.
Jeffcott rotor
The Jeffcott rotor (named after Henry Homan Jeffcott), also known as the
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.
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.
Gustaf de Laval
Karl Gustaf Patrik de Laval (; 9 May 1845 – 2 February 1913) was a Swedish engineer and inventor who made important contributions to the design of steam turbines and centrifugal separation machinery for dairy.
Life
Gustaf de Laval was born a ...
, 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
August Otto Föppl (25 January 1854 – 12 August 1924) was a professor of Technical Mechanics and Graphical Statics at the Technical University of Munich, Germany. He is credited with introducing the Föppl–Klammer theory and the 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
Nils Otto Myklestad and
M. A. Prohl 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
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
.
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.
Software
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.
Rotordynamic specific codes:
* Dynamics R4 (Alfa-Tranzit Co. Ltd) - Commercial software developed for design and analysis of spatial systems
* AxSTREAM RotorDynamics, (SoftInWay) - Integrated Software platform for Rotor Dynamics, capable of lateral, torsional, and axial rotor dynamics for all widely used rotor types using the Finite Element Method on either beam or 2D-axisymmetric elements, and is capable of being automated.
* Rotortest, (LAMAR -
University of Campinas
The State University of Campinas ( pt, Universidade Estadual de Campinas), commonly called Unicamp, is a public research university in the state of São Paulo, Brazil. Unicamp is consistently ranked among the top universities in Brazil and Latin ...
) - Finite Element Method based software, including different types of bearing solver. Developed by LAMAR (Laboratory of Rotating Machinery) - Unicamp (University of Campinas).
*
SAMCEF SAMCEF is a finite element analysis (FEA) software package dedicated to mechanical virtual prototyping. SAMCEF development started in 1965 at the Aerospace Laboratory of University of Liège. It was developed and sold by SAMTECH, a Belgian compan ...
ROTOR - Software Platform for Rotors Simulation (LMS Samtech,A Siemens Business)
* MADYN (Consulting engineers Klement) - Commercial combined finite element lateral, torsional, axial and coupled solver for multiple rotors and gears, including foundation and housing.
* MADYN 2000 (DELTA JS Inc.) - Commercial combined finite element (3D Timoshenko beam) lateral, torsional, axial and coupled solver for multiple rotors and gears, foundations and casings (capability to import transfer functions and state space matrices from other sources), various bearings (fluid film, spring damper, magnetic, rolling element)
* iSTRDYN (DynaTech Software LLC) - Commercial 2-D Axis-symmetric finite element solver
* FEMRDYN (DynaTech Engineering, Inc.) - Commercial 1-D Axis-symmetric finite element solver
* DyRoBeS (Eigen Technologies, Inc.) - Commercial 1-D beam element solver
* RIMAP (RITEC) - Commercial 1-D beam element solver
* XLRotor (Rotating Machinery Analysis, Inc.) - Commercial 1-D beam element solver, including magnetic bearing control systems and coupled lateral-torsional analysis. A powerful, fast and easy to use tool for rotor dynamic modeling and analysis using Excel spreadsheets. Readily automated with VBA macros, plus a plugin for 3D CAD software.
* ARMD
Rotor Bearing Technology & Software, Inc. - Commercial FEA-based software for rotordynamics, multi-branch torsional vibration, fluid-film bearings (hydrodynamic, hydrostatic, and hybrid) design, optimization, and performance evaluation, that is used worldwide by researchers, OEMs and end-users across all industries.
* XLTRC2 (
Texas A&M
Texas A&M University (Texas A&M, A&M, or TAMU) is a public, land-grant, research university in College Station, Texas. It was founded in 1876 and became the flagship institution of the Texas A&M University System in 1948. As of late 2021, T ...
) - Academic 1-D beam element solver
* ComboRotor (
University of Virginia
The University of Virginia (UVA) is a Public university#United States, public research university in Charlottesville, Virginia. Founded in 1819 by Thomas Jefferson, the university is ranked among the top academic institutions in the United S ...
) - Combined finite element lateral, torsional, axial solver for multiple rotors evaluating critical speeds, stability and unbalance response extensively verified by industrial use
* MESWIR (Institute of Fluid-Flow Machinery,
Polish Academy of Sciences
The Polish Academy of Sciences ( pl, Polska Akademia Nauk, PAN) is a Polish state-sponsored institution of higher learning. Headquartered in Warsaw, it is responsible for spearheading the development of science across the country by a society of ...
) - Academic computer code package for analysis of rotor-bearing systems within the linear and non-linear range
* RoDAP (D&M Technology) - Commercial lateral, torsional, axial and coupled solver for multiple rotors, gears and flexible disks(HDD)
* ROTORINSA (ROTORINSA) - Commercial finite element software developed by a French engineering school (INSA-Lyon) for analysis of steady-state dynamic behavior of rotors in bending.
*
COMSOL Multiphysics
COMSOL Multiphysics is a finite element analysis, solver, and simulation software package for various physics and engineering applications, especially coupled phenomena and multiphysics. The software facilitates conventional physics-based us ...
, Rotordynamics Module add-on (Rotordynamics Module)
* RAPPID - (Rotordynamics-Seal Research)
Commercial finite element based software library (3D solid and beam elements) including rotordynamic coefficient solvers
See also
*
Axle
An axle or axletree is a central shaft for a rotating wheel or gear. On wheeled vehicles, the axle may be fixed to the wheels, rotating with them, or fixed to the vehicle, with the wheels rotating around the axle. In the former case, bearing ...
*
Balancing machine
A balancing machine is a measuring tool used for balancing rotating machine parts such as rotors for electric motors, fans, turbines, disc brakes, disc drives, propellers and pumps. The machine usually consists of two rigid pedestals, with suspe ...
*
Bearing (mechanical)
A bearing is a machine element that constrains relative motion to only the desired motion, and reduces friction between moving parts. The design of the bearing may, for example, provide for free linear movement of the moving part or for free ro ...
*
Driveshaft
A drive shaft, driveshaft, driving shaft, tailshaft (Australian English), propeller shaft (prop shaft), or Cardan shaft (after Girolamo Cardano) is a component for transmitting mechanical power and torque and rotation, usually used to connec ...
*
Exoskeletal engine
*
Magnetic bearing
A magnetic bearing is a type of bearing that supports a load using magnetic levitation. Magnetic bearings support moving parts without physical contact. For instance, they are able to levitate a rotating shaft and permit relative motion with ve ...
*
Turbine
A turbine ( or ) (from the Greek , ''tyrbē'', or Latin ''turbo'', meaning vortex) is a rotary mechanical device that extracts energy from a fluid flow and converts it into useful work. The work produced by a turbine can be used for generating e ...
References
* uses DyRoBeS
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Ganeriwala, S., Mohsen N (2008). ''Rotordynamic Analysis using XLRotor.'
SQI03-02800-0811
Notes
{{reflist
External links
Rotordynamic Analysis using XLRotorGateway to technical literature on Rotordynamics
Dynamics (mechanics)