HOME

TheInfoList



OR:

Rotordynamics (or 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 A jet engine is a type of reaction engine, discharging a fast-moving jet (fluid), jet of heated gas (usually air) that generates thrust by jet propulsion. While this broad definition may include Rocket engine, rocket, Pump-jet, water jet, and ...
and
steam turbine A steam turbine or steam turbine engine is a machine or heat engine that extracts thermal energy from pressurized steam and uses it to do mechanical work utilising a rotating output shaft. Its modern manifestation was invented by Sir Charles Par ...
s to auto engines and computer
disk storage Disc or disk may refer to: * Disk (mathematics) In geometry, a disk (Spelling of disc, also spelled disc) is the region in a plane (geometry), plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitut ...
. 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 (such as bacterial flagella or ATP synthase). Energy flows through a stator to or from the rotat ...
. 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 approaches ...
. This amplitude is commonly excited by imbalance of the rotating structure; everyday examples include
engine balance Engine balance refers to how the inertial forces produced by moving parts in an internal combustion engine or steam engine are neutralised with counterweights and Balance shaft#Overview, balance shafts, to prevent unpleasant and potentially dam ...
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 (wheel), rim) on which it is mounted. When the wheel rotates, asymmetries in its mass ...
. 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 ot ...
occurs. In addition to this,
turbomachinery Turbomachinery, in mechanical engineering, describes machines that transfer energy between a Rotor (electric), rotor and a fluid, including both turbines and gas compressor, compressors. While a turbine transfers energy from a fluid to a rotor, ...
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, measured in terms of '' eigenfrequency'', is the rate at which an oscillatory system tends to oscillate in the absence of disturbance. A foundational example pertains to simple harmonic oscillators, such as an idealized spring ...
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 is a phenomenon that occurs when an object or system is subjected to an external force or vibration whose frequency matches a resonant frequency (or resonance frequency) of the system, defined as a frequency that generates a maximu ...
. 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 models), equations obtained from solving models numerically ( Rayleigh–Ritz method) and finally from the
finite element method 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 tran ...
(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. More specifically, the equations of motion describe the behavior of a physical system as a set of mathem ...
, in generalized
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
form, for an axially symmetric rotor rotating at a constant spin speed is \mathbf \ddot(t) + (\mathbf + \mathbf) \dot(t) + (\mathbf + \mathbf) \mathbf(t) = \mathbf(t) where: : is the
symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
mass matrix; : is the symmetric damping matrix; : is the skew-symmetric gyroscopic matrix: : is the symmetric bearing or seal stiffness matrix; : is the gyroscopic matrix of deflection for inclusion of e.g., centrifugal elements; : is the generalized coordinates of the rotor in inertial coordinates; : is a forcing function, usually including the unbalance. The gyroscopic matrix is proportional to spin speed . 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 is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...
which are spin speed dependent. Engineering specialists in this field rely on the
Campbell Diagram A Campbell diagram plot represents a system's response spectrum as a function of its oscillation regime. It is named for Wilfred Campbell, who introduced the concept. It is also called an interference diagram. In rotordynamics In Rotordynamics, ...
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.


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. A Jeffcott rotor consists of a flexible, massless, uniform shaft mounted on two flexible bearings equidistant from a massive disk rigidly attached to the shaft. The simplest form of the rotor constrains the disk to a plane orthogonal to the axis of rotation. This limits the rotor's response to lateral vibration only. If the disk is perfectly balanced (i.e., its geometric center and center of mass are coincident), then the rotor is analogous to a single-degree-of-freedom undamped
oscillator Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
under free vibration. If there is some radial distance between the geometric center and center of mass, then the rotor is unbalanced, which produced a force proportional to the disk's mass, , the distance between the two centers (eccentricity, ) and the disk's spin speed, . After calculating the equivalent stiffness, , of the system, we can create the following second-order linear ordinary differential equation that describes the radial deflection of the disk from the rotor centerline. m \mathbb + k \mathbb = m \varepsilon \Omega^2 \sin(\Omega t) If we were to graph the radial response, we would see a sine wave with angular frequency \Omega/2\pi. This lateral oscillation is called 'whirl', and in this case, is highly dependent upon spin speed. Not only does the spin speed influence the amplitude of the forcing function, it can also produce dynamic amplification near the system's natural frequency. While the Jeffcott rotor is a useful tool for introducing rotordynamic concepts, it is important to note that it is a mathematical
idealization Psychoanalytic theory posits that an individual unable to integrate difficult feelings mobilizes specific defenses to overcome these feelings, which the individual perceives to be unbearable. The defense that effects (brings about) this process i ...
that only loosely approximates the behavior of real-world rotors.


Campbell diagram

The
Campbell diagram A Campbell diagram plot represents a system's response spectrum as a function of its oscillation regime. It is named for Wilfred Campbell, who introduced the concept. It is also called an interference diagram. In rotordynamics In Rotordynamics, ...
, 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 approaches ...
.


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 at ...
, 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 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 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 tran ...
. 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.


See also

*
Axle An axle or axletree is a central shaft for a rotation, rotating wheel and axle, 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 ...
* Balancing machine *
Bearing (mechanical) A ball bearing 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 pa ...
*
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, torque, and rotation, usually used to connect ...
*
Exoskeletal engine The exoskeletal engine (ESE) is a concept in turbomachinery design. Current gas turbine engines have central rotating shafts and fan-discs and are constructed mostly from heavy metals. They require lubricated bearings and need extensive cooling fo ...
*
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 v ...
*
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 can be used for generating electrical ...


References

* uses DyRoBeS * * * * * * * * * * * * * * * * * Ganeriwala, S., Mohsen N (2008). ''Rotordynamic Analysis using XLRotor.'
SQI03-02800-0811


Notes

{{reflist


External links


Rotordynamic Analysis using XLRotor

Gateway to technical literature on Rotordynamics
Dynamics (mechanics) Rotation