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# Critical speed

About: Critical speed is a(n) research topic. Over the lifetime, 2764 publication(s) have been published within this topic receiving 31365 citation(s).

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TL;DR: In this paper, a crack in a structural member introduces a local flexibility that affects its vibration response, and the crack will open and close in time depending on the rotation and vibration amplitude.

Abstract: The presence of a crack in a structural member introduces a local flexibility that affects its vibration response. Moreover, the crack will open and close in time depending on the rotation and vibration amplitude. In this case the system is nonlinear. Furthermore, if general motion is considered, the local stiffness matrix description of the cracked section of the shaft leads to a coupled system, while for an uncracked shaft the system is decoupled. This means that the crack introduces new harmonics in the spectrum. In fact, in addition to the second harmonic of rotation and the subharmonic of the critical speed, two more families of harmonics are observed: 1. (1) higher harmonics of the rotating speed due to the nonlinearity of the closing crack, and 2. (2) longitudinal and torsional harmonics are present in the start-up lateral vibration spectrum due to the coupling. Over 500 papers on the subject were published in the past 10 yrs. A wealth of analytical, numerical and experimental investigations now exists. However, a consistent cracked bar vibration theory is yet to be developed. There are still many unanswered questions, especially in the area of closing cracks in rotating shafts.

996 citations

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01 Jan 2005

TL;DR: In this paper, the authors present the results of almost 30 years of work in the field of rotordynamics, which includes research, teaching, writing computer codes, and consulting.

Abstract: This book is the result of almost 30 years of work in the field of rotordynamics, which includes research, teaching, writing computer codes, and consulting. It is the outcome of an interdisciplinary research team that operated, and still operates, in the Mechanics Department and in the Interdepartmental Mechatronics Laboratory of Politecnico di Torino. The aim is mostly to write in a systematic way what has been the subject of a number of research papers, in such a way to give a consistent picture of the dynamic behavior of rotating machinery. As the title implies, this book is an attempt (only the reader can judge whether it is successful) to go beyond what is usually referred to as rotordynamics. The aim is that of dealing with the dynamic behavior of systems having in common the feature of rotating. This definition includes obviously those systems, like transmission shafts, turbine rotors, and gyroscopes, which are studied by rotordynamics, but also systems such as rotating blades (like in helicopter rotors) or flexible spinning spacecraft. Although rotordynamics usually deals only with the lateral behavior of rotors, some mention is made here also to torsional and axial vibration or to cases in which it is impossible to distinguish between them. However, the author imposed a limitation: No mention will be made of the dynamics of machines containing reciprocating parts, such as a crankshaft-connecting rod-piston mechanism. This arbitrary decision is based on the grounds that their vibration (mainly torsional vibration, but also axial and lateral vibration) is a very specialized topic, dealt with in many handbooks and textbooks and, above all, that to include it would have meant either to give a very insubstantial account or to double the size of the book. Another area in which a decision about where to stop was needed is controlled rotors. A thorough study of the dynamics of many controlled rotors, like those running on active magnetic bearings or supplied with active dampers, would have implied a detailed study of their control systems (hardware and, in case of digital systems, software) sensors and actuators (with the critical issue of the power amplifiers). As is typical of mechatronic systems, only an integrated and interdisciplinary approach allows us to exploit the advantages of the potentialities modern technology has opened. As this would have lead too far from the main topics of this book, these areas will be touched only marginally. The text is structured in two parts. The first one deals with what could be defined as classic or basic rotordynamics. The contents are basically well consolidated, although some incorrect statements can be found even in recent papers published on well-known journals. The basic assumptions are linearity, steady state operation, and at least some degree of axial symmetry. The second part, containing topics that are usually considered as specialized aspects of rotordynamics, could be titled advanced rotordynamics. The mentioned assumptions are dropped, and more detailed models are built for rotors departing from the classic configurations studied in rotordynamics. The contents of this part are more research topics than consolidated applications. The contents and the credits for the various chapters are the following: Chapter 1: Introduction. The basic concepts, graphical representation, and methods of rotordynamics are illustrated in a qualitative way. The expert reader, although familiar with these concepts, should not skip it altogether because the basic notation and the viewpoint that will be followed in the whole text are described. Part 1: Basic topics Chapter 2: Jeffcott rotor. The so-called Jeffcott rotor is the simplest rotor model that can be conceived. Although unable to account for some typical phenomena linked with rotordynamics, like gyroscopic effect or centrifugal stiffening, it allows us to gain a good insight into the peculiarities of rotating systems. In particular, it is essential for understanding the role of damping in rotordynamics. The topics dealt with are as a whole standard, but the part on nonsynchronous damping, is less common. Chapter 3: Model with four degrees of freedom: Gyroscopic effect. A simple model in which a rigid body is substituted for the point mass of the Jeffcott rotor is then studied, to allow the study of gyroscopic effects. This model is representative for the behavior of any rigid rotor on compliant bearings and allows us to define a modal gyroscopic system, on which modal decomposition of rotors can be based under some assumptions. Chapter 4: Discrete multi-degrees-of-freedom rotors. The lateral behavior of a flexible rotor modeled as a discrete parameter beamlike (1-D approach) system is then studied. Older approaches, like the transfer matrices methods, are dealt with together with more modern ones, like the finite element method (FEM). Chapter 5: Continuous systems: Transmission shafts. A short account on modeling simple rotors as continuous system is then included. This chapter can be considered more of academic rather than of practical relevance. Chapter 6: Anisotropy of rotors or supports. If either the rotor or the stator are not isotropic, it is still possible to obtain a closed-form solution for the linearized steady-state dynamics. Such systems are studied with particular reference to the backward whirling caused by unbalance in isotropic rotors on asymmetric supports and to the instability ranges of nonsymmetric rotors on isotropic supports. Chapter 7: Torsional and axial dynamics. The axial and torsional dynamics of rotors is briefly dealt with. Considering that the torsional and axial behavior is unaffected by the rotation of the system (at least if the basic assumptions of linearity and small displacements are made), just a brief account is reported. Chapter 8: Rotor-bearings interaction. The interaction between the behavior of the rotor and of the bearing is a complex subject, mainly because of the nonlinear behavior of the latter. The approach here followed is the classic one: The nonlinearity of the bearings is accounted for in computing their working conditions, and then the dynamic behavior is linearized assuming small displacements about the static equilibrium position (at speed). Rolling elements and lubricated and magnetic bearings are dealt with. Part 2: Advanced topics Chapter 9: Anisotropy of rotors and supports. The assumption that either the stator or the rotor is isotropic is dropped. No closed-form solution is any more possible, although a truncated series solution can be attempted. Chapter 10: Nonlinear rotordynamics. Here another assumption, that of linearity, is dropped. The phenomena typical of nonlinear systems, like jumps and even chaotic behavior are discussed. Chapter 11: Nonstationary rotordynamics. The spin speed is no more assumed to be constant, or other parameters, like unbalance, are allowed to change. In particular, the acceleration of the rotor through a critical speed and the occurrence of a blade loss are dealt with in detail. Chapter 12: Dynamic behavior of free rotors. Unconstrained rotating objects, like spinning celestial bodies or spacecraft, can be considered as rotors. The main aim of this section is to show that the assumption of constant angular momentum, typical of the dynamic study of free rotors, and that of constant angular velocity, typical of classic rotordynamics, coincide when the small displacement and rotations assumptions is made, so that the first can be approached with the methods of the latter. Chapter 13: Dynamics of rotating beams and blades. The effect of rotation, about an axis perpendicular to their longitudinal axis, on the dynamic behavior of beams and the blades-rotor interaction is studied using simple models. The well-known phenomena related to propeller and helicopter rotors' instability are dealt with, as well as other less-known phenomena regarding the effects of blade damping on the stability of a bladed rotor. Chapter 14: Dynamics of rotating discs and rings. Turbine and compressor discs are assumed, in classic rotordynamics, to behave as rigid bodies. In this chapter, this assumption is dropped and the effects of the flexibility of the discs are dealt with using simple models, starting from that introduced about 80 years ago by Southwell. Chapter 15: Three-dimensional modeling of rotors. This chapter deals with numerical modeling, mostly based on the FEM, of complex rotors. The topics dealt with in Chapters 13 and 14 using simplified models are here treated with the aim of building more accurate models, yielding precise quantitative results. Chapter 16: Dynamics of controlled rotors. Active vibration control is increasingly applied to rotors, either together with the use of active magnetic suspension or with techniques using active dampers or the control of more or less conventional bearings. As already stated, no attempt in modeling in detail the control, sensor or actuator dynamics is done, because it would lead too far from the central topics of this book. Appendix A: Vectors, matrices, and equations of motion. Some basic topics of system dynamics, particularly for the peculiar aspects linked with rotating systems, are summarized in this appendix. Appendix B: An outline on rotor balancing. As many very good books have been written on rotor balancing, only a short account on the basic topics are dealt with. Appendix E: Bibliography. Some of the books specifically devoted to rotordynamics are listed in chronological order. A CD-ROM comes with this book. It contains a simplified version of the DYNROT code and two short videos

617 citations

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TL;DR: In this paper, it is shown that large dynamic amplifications appear in the dynamic response of the rail/embankment/ground system as the train speed approaches an apparently critical value.

Abstract: Results from instrumented test runs with a high-speed train on a soft soil site in Sweden are presented. It is shown that large dynamic amplifications appear in the dynamic response of the rail/embankment/ground system as the train speed approaches an apparently critical value. The measured dynamic response is analyzed in detail, and it is shown that the critical speed is controlled by the minimum phase velocity of the first Rayleigh mode of the soil and embankment profile at the site. Moreover, it is shown that the critical speed and the amount of dynamic amplification also depend on a coincidence between characteristic wavelengths for the site and the distances between bogies and axles in the train. The displacement response is found to consist of a speed-independent portion in quasi-static equilibrium with the train loads and a dynamic portion representing freely propagating Rayleigh waves. An efficient computer code for the prediction of ground response to high-speed trains has been developed and its ability to reproduce the observed behaviour is demonstrated.

335 citations

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TL;DR: In this paper, a perturbation theory for the near-modal free vibration of a general gyroscopic system with weakly nonlinear stiffness and/or dissipation is derived through the asymptotic method of Krylov, Bogoliubov, and Mitropolsky.

Abstract: Free non-linear vibration of an axially moving, elastic, tensioned beam is analyzed over the sub- and supercritical transport speed ranges. The pattern of equilibria is analogous to that of Euler column buckling and consists of the straight configuration and of non-trivial solutions that bifurcate with speed. The governing equations for finite local motion about the trivial equilibrium and for motion about each bifurcated solution are cast in the standard form of continuous gyroscopic systems. A perturbation theory for the near-modal free vibration of a general gyroscopic system with weakly non-linear stiffness and/or dissipation is derived through the asymptotic method of Krylov, Bogoliubov, and Mitropolsky. The method is subsequently specialized to non-linear vibration of a traveling beam, and of a traveling string in the limit of vanishing flexural rigidity. The contribution of non-linear stiffness to the response increases with subcritical speed, grows most rapidly near the critical speed, and can be several times greater for a translating beam than for one that is not translating. In the supercritical speed range, asymmetry of the non-linear stiffness distribution biases finite-amplitude vibration toward the straight configuration and lowers the effective modal stiffness. The linear vibration theory underestimates stability in the subcritical range, overestimates it for supercritical speeds, and is most limited in the near-critical regime.

304 citations

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24 Oct 2000

TL;DR: The RDA Code for Lateral Rotor Vibration analysis is described in this paper, where the authors present a detailed overview of the RDA Software Insights into Linear LRVs (SLRVs).

Abstract: Part I: Primer on Rotor Vibration Vibration Concepts and Methods One-Degree-of-Freedom Model Multi-DOF Models Modes, Excitation, and Stability of Multi-DOF Models Lateral Rotor Vibration Analysis Models Simple Linear Models Formulations for RDA Software Insights into Linear LRVs Nonlinear Effects in Rotor Dynamical Systems Torsional Rotor Vibration Analysis Models Rotor-Based Spinning Reference Frames Single Uncoupled Rotor Coupled Rotors Semidefinite Systems Part II: Rotor Dynamic Analyses RDA Code for Lateral Rotor Vibration Analyses Unbalance Steady-State Response Computations Instability Self-Excited-Vibration Threshold Computations Additional Sample Problems Bearing and Seal Rotor Dynamics Liquid-Lubricated Fluid-Film Journal Bearings Experiments to Measure Dynamic Coefficients Annular Seals Rolling Contact Bearings Squeeze-Film Dampers Magnetic Bearings Compliance Surface Foil Gas Bearings Turbo-Machinery Impeller and Blade Effects Centrifugal Pumps Centrifugal Compressors High-Pressure Steam Turbines and Gas Turbines Axial Flow Compressors Part III Monitoring and Diagnostics Rotor Vibration Measurement and Acquisition Introduction to Monitoring and Diagnostics Measured Vibration Signals and Associated Sensors Vibration Data Acquisition Signal Conditioning Vibration Severity Guidelines Casing and Bearing Cap Vibration Displacement Guidelines Standards, Guidelines, and Acceptance Criteria Shaft Displacement Criteria Signal Analysis and Identification of Vibration Causes Vibration Trending and Baselines FFT Spectrum Rotor Orbit Trajectories Bode, Polar, and Spectrum Cascade Plots Wavelet Analysis Tools Chaos Analysis Tools Symptoms and Identification of Vibration Causes Part IV Trouble-Shooting Case Studies Forced Vibration and Critical Speed Case Studies HP Steam Turbine Passage through First Critical Speed HP-IP Turbine Second Critical Speed through Power Cycling Boiler Feed Pumps: Critical Speeds at Operating Speed Nuclear Feed Water Pump Cyclic Thermal Rotor Bow Power Plant Boiler Circulating Pumps Nuclear Plant Cooling Tower Circulating Pump Resonance Generator Exciter Collector Shaft Critical Speeds Self-Excited Rotor Vibration Case Studies Swirl Brakes Cure Steam Whirl in a 1300 MW Unit Bearing Unloaded by Nozzle Forces Allows Steam Whirl Misalignment Causes Oil Whip/Steam Whirl "Duet" Additional Rotor Vibration Cases and Topics Vertical Rotor Machines Vector Turning from Synchronously Modulated Rubs Air Preheater Drive Structural Resonances Aircraft Auxiliary Power Unit Commutator Vibration-Caused Uneven Wear Impact Tests for Vibration Problem Diagnoses Bearing Looseness Effects Tilting-Pad versus Fixed-Surface Journal Bearings Base-Motion Excitations from Earthquake and Shock Parametric Excitation: Nonaxisymmetric Shaft Stiffness Rotor Balancing Index

212 citations