Rotordynamics

As a result of continuing demand for increased performance, modern generators are sometimes designed to operate near critical speed, which causes difficulty in maintaining the rotor balance required to ensure acceptable vibration levels. Forced vibration due to the external load acting on the shaft line is not considered in evaluation of critical speeds of multi-section shaft. The importance of the current work carried out is to evaluate the critical speed through an automated process. By this process any configuration of rotor can be used to evaluate the critical speed of multi-section rotor. Thus for evaluation of critical speed and to study the rotor behavior in dynamic condition, a script is generated in order to automate the process, thereby reducing the time consumed & increasing the accuracy. The uniqueness of this methodology adopted is that with the help of one script the input like diameter, length of each step of rotor, load acting on the rotor is accessed as variable into the script. The script is generated in ANSYS using Ansys Programming Design Language (APDL). The generated script results are correlated with theoretical results and the measured results, which are well within the acceptable limits. Important conclusion of the study is achieving complete automation for evaluating the critical speed by considering external load & for any configuration or combination of rotors.

The propellant used in space programs creates exclusively three ecological concerns: Ground-based Impacts which vary from groundwater contagion to explosions caused by inconvenient management of fuels. Atmospheric Impacts are generally due to the interface of propellant exhaust with the atmosphere. Biological Impacts include toxicity and corrosiveness of propellants. This leads to the emergence of Cryogenic and Semi Cryogenic engines where the main propellants are Liquid Hydrogen and Liquid Oxygen (very effective in controlling above mentioned three environmental concerns). The operation necessities designate a need for reliable and long-life LH 2 & LOX turbopumps. Existing turbopumps use rolling component bearings which restrict turbopump life. Foil Gas Bearings (FGB) can be effectively used in high-speed turbomachines to attain noteworthy consistency. The present work reports how FGB can successfully replace the conventional roller bearings and prove to be a very important contribution towards Green Propulsion Technology. FGB offers elimination of oil lubrication system (an environmental concern) resulting into condensed weight and improved temperature competence. Using FGB to cryo turbopumps would lengthen turbopump life well ahead of operation necessities. This work also focuses on how Foil Gas Bearing technology can be a great prospect for Indian Space Programmes.

Rotating machinery acceptance parameters in terms of vibration and protection is one of the key considerations in detailed engineering and design process. For these activities, API standards are used normally in oil and gas industries. For centrifugal pumps, API 610 and 670 may mandate too stringent requirement for particular type of pumps in terms of acceptance criteria irrespective to actual duty of pump. There is a growing desire for using hydrodynamic bearings irrespective to energy level of pump in a general believe that they have virtually infinite life span. The acceptance criteria during shop tests as mandated by API 610 has not recognized the bearing and foundation flexibility. This is a potential grey area for machinery engineers from end user/contractor engaged in detailed engineering. The objective of this article is to provide the insights of the aspect.

Conventional rotordynamic analyses generally simplify the rotor, neglecting detailed geometrical characteristics. However, in modern rotating machines, rotors consist of multiple complex-shaped parts that are usually non-axisymmetric with preloads to ensure the assembly. These effects may significantly affect rotordynamic behavior of high-performance rotating machinery. The present study aims to take them into account in rotordynamic analyses, by presenting an efficient rotordynamic stability approach for non-axisymmetric rotor-bearing systems with complex shapes using threedimensional solid finite elements. The 10-node quadratic tetrahedron element is used for the finite element formulation of the rotor. A rotor-bearing system, matrix differential equation is derived in the rotor-fixed coordinate system. The system matrices are reduced by using Guyan reduction. The current study utilizes the Floquet theory to determine the stability of solutions for parametrically excited rotor-bearing systems. Computational efficiency is improved by discretization and parallelization, taking advantage of the discretized Monodromy matrix of Hsu's method. The method is

Rolling bearings are one of the most important uproar and vibration generator into mechanical constructions. According to that, it is very important to know ascendancy of rolling bearing construction to main parameters which determine level of uproar and vibrations produced inside machines. This paper present new mathematical model of dynamic behavior of rigid rotor inside the rolling bearing. The new model is experimentally validated on a vibration test of rotor.

This paper presents a novel topology for enhanced vibration sensing in which wireless MEMS accelerometers embedded within a hollow rotor measure vibration in a synchronously rotating frame of reference. Theoretical relations between rotor-embedded accelerometer signals and the vibration of the rotor in an inertial reference frame are derived. It is thereby shown that functionality as a virtual stator-mounted displacement transducer can be achieved through appropriate signal processing. Experimental tests on a prototype rotor confirm that both magnitude and phase information of synchronous vibration can be measured directly without additional stator-mounted key-phasor sensors. Displacement amplitudes calculated from accelerometer signals will become erroneous at low rotational speeds due to accelerometer zero-g offsets, hence a corrective procedure is introduced. Impact tests are also undertaken to examine the ability of the internal accelerometers to measure transient vibration. A further capability is demonstrated, whereby the accelerometer signals are used to measure rotational speed of the rotor by analysing the signal component due to gravity. The study highlights the extended functionality afforded by internal accelerometers and demonstrates the feasibility of internal sensor topologies, which can provide improved observability of rotor vibration at externally inaccessible rotor locations.

Electrodynamic bearings (EDBs) are passive magnetic bearings that exploit the interaction between eddy currents developed in a rotating conductor and a static magnetic field to generate forces. Similar to other types of magnetic suspensions, EDBs provide contactless support, thus avoiding problems with lubrication, friction and wear. Electrodynamic bearings have also drawbacks such as the difficulty in insuring a stable levitation in a wide speed range. The paper presents a solution where the EDBs are coupled with active magnetic dampers (AMDs) to guarantee a stable levitation.

In this paper, the complementary role of Finite Element Analysis (FEA) and Modal Analysis is studied using ANSYS Mechanical APDL. Primarily, two models (2D and 3D) of the flywheel are designed. Then, Finite Element (FE) static-stress analysis of the model was carried out, and it is found that the maximum von Mises stress (403 MPa) on the flywheel is lower than the yield stress (850 MPa) level, i.e. the design of the flywheel is safe. Afterwards, the Modal Analysis of the 3D model is carried out and six mode shapes are obtained at six different natural frequencies. Furthermore, Campbell diagram is plotted and two natural frequencies are identified. Moreover, mode shapes are plotted at the two critical speeds corresponding to their respective damped natural frequencies. The mode shapes are found to be stable, and therefore it can be remarked that the flywheel model is safe to operate within 12000 rpm. Finally, the performance of ANSYS models is determined by comparing analytical solut...

Les instabilites generees par frottement sont responsables des divers bruits tels que le crissement, le sifflement ou le broutement ... Pour modeliser et comprendre ce phenomene d'instabilites, les analyses temporelle et frequentielle sont utilisees sur un systeme modele constitue de deux poutres en contact. Dans l'analyse frequentielle, lineaire, l'instabilite se manifeste par la coalescence de deux modes propres du systeme. Dans l'analyse temporelle, qui tient compte de l'aspect non lineaire d'un contact frottant, l'instabilite est caracterisee par des zones d'adherence ou de decollement qui apparaissent au niveau de la surface de contact. Les resultats issus des deux analyses sont coherents et complementaires, malgre quelques differences de prediction. Une validation experimentale a ete effectuee et montre une bonne correlation entre les resultats numeriques temporels et experimentaux. On met en evidence la pertinence de l'analyse temporelle da...

Ce papier traite de l'étude harmonique d'un modèle réduit d'aubage obtenu par projection modale et soumis à des non-linéarités géométriques. Les effets non-linéaires réduits sont supposés polynomiaux et déterminés à l'aide d'une procédure d'évaluation des raideurs non-linéaires (méthode STEP). Les réponses stationnaires dans les cas forcé et libre sont obtenues grâce à la méthode de l'équilibrage harmonique. Une résonance interne de type 1 : 2 est déterminée. Celle-ci traduit un couplage de modes et apparait au niveau des points de bifurcation de type point d'embranchement. Mots clés -méthode de l'équilibrage harmonique, résonance interne, stiffness evaluation procedure.

The present study aims to predict the dynamic stability of a turbofan engine submitted to light contacts between fan blades tips and their surrounding casing. The study is based on a hybrid model which introduces a simplified bladed wheel and a flexible suspended casing to a rotor-shaft model. Inertial coupling is considered between blades and shaft dofs and an elastic link is introduced between the shaft and the suspension dofs of the casing. A 3D contact formulation has been implemented to recover accuracy in the contacts detection and the application of contact forces; it considers the 3D model kinematic and introduces the local geometry of the contact area. The model behavior with blade-tocasing contacts is analyzed through two approaches: the first one assumes permanent contacts while the other one allows intermittent contacts. Good agreement is obtained between the two types of results. The results highlight the importance of couplings in the initiation of unstable phenomena and the relevance of the 3D contact formulation in predicting the stability of the system.

A mesoscale turbocompressor spinning above 500,000 RPM is evolutionary technology for micro turbochargers, turbo blowers, turbo compressors, micro-gas turbines, auxiliary power units, etc for automotive, aerospace, and fuel cell industries. Objectives of this work are: 1) ...

Unbalanced magnetic pull (UMP) resulting from air-gap eccentricity can present a potential risk to the lifetime and dynamic stability of high-speed electrical machines. Nevertheless, a method to identify the effects of UMP in actual industrial machines has not yet been sufficiently developed. In this paper, methods for analysis and experimental verification of UMP effects are studied using a high-speed two-pole induction generator supported by active magnetic bearings (AMBs) as a case example. The UMP force is calculated using a semi-analytical model that combines an analytical model with a correction factor obtained from finite element analysis (FEA) results. Using this model, the characteristics of time-variant UMP that are related to the effects of UMP on rotordynamics are investigated. Coefficients for the rotor-bearing simulation model are identified using a detailed CAD model and experimental modal analysis data. Linearized coefficients of AMBs are identified based on the rigid body whirling mode of the rotor. Then, UMP effects are investigated by conducting a time-step rotordynamic simulation in the mixed eccentricity condition, and the results are verified by comparing them with the vibration measurement results during ramp-down operation of the test machine. Results show two main effects produced by UMP on the rotordynamics of induction machines, namely reduction in the rotor natural frequency and additional vibration caused by twice the supply frequency excitation, thus confirming that the proposed semi-analytical UMP model is suitable for the rotordynamics simulation and achieves a high accuracy with efficient computation. INDEX TERMS Active magnetic bearings, air-gap eccentricity, electromagnetic forces, induction motors, rotordynamics, unbalanced magnetic pull, vibrations.

In large rotor-bearing systems, the rolling element bearings act as a considerable source of subcritical vibration excitation. Simulation of such rotor bearing systems contains major sources of uncertainty contributing to the excitation, namely the roundness profile of the bearing inner ring and the clearance of the bearing. In the present study, a simulation approach was prepared to investigate carefully the effect of varying roundness profile and clearance on the subcritical vibration excitation. The FEM-based rotor-bearing system simulation model included a detailed description of the bearings and asymmetricity of the rotor. The simulation results were compared to measured responses for validation. The results suggest that the simulation model was able to capture the response of the rotor within a reasonable accuracy compared to the measured responses. The bearing clearance was observed to have a major effect on the subcritical resonance response amplitudes. In addition, the simulation model confirmed that the resonances of the 3rd and 4th harmonic vibration components in addition to the well-known 2nd harmonic resonance (half-critical resonance) can be significantly high and should thus be taken into account already in the design phase of large subcritical rotors.

In this study, different numerical models are computed and compared with particle image velocimetry (PIV) measurement for wake correlation of a 4.5 m of diameter wind turbine rotor. The collaborative European wind turbine MEXICO project is carried out in three commonly experimental defined test cases at wind speeds of 10, 15 and 24 m.s-1. To discuss the rotor near wake, a laminar Navier-Stokes approach and a Reynolds averaged Navier-Stokes turbulent model both coupled with an actuator disc (AD) technique are computed, then compared with a direct model from the literature called TAU/DM as a full rotor technique. The actuator disc momentums are calculated using user defined functions (UDF) in FLUENT introduced as UDF/AD technique in this paper. The results are discussed in detail and compared with PIV detailed measurements.

Rotordynamics is a very challenging field because of machine complexities. Many internal and external factors contribute toward change in the structural dynamic characteristics. One of these factors is broad-band high-vibration amplitudes. In this article, a similar high vibration issue on a gas turbine is investigated using bode, orbit, and shaft centerline plots. Data from proximity probes installed on turbine generator system are captured and analyzed against any factor contributing toward high vibration issue. Fish bone diagram was used for root cause investigation. Main components investigated for the root cause of high vibration issue include generator rotor and casing. Rotor behavior has been examined by capturing orbit and shaft centerline diagrams, whereas casing contribution has been investigated by conducting operating deflection shape analysis. A comparison is drawn between a machine suffering from high vibration issue and a normal machine. Resonance was identified as th...

Bladed disks of engine rotors usually operate at harsh conditions of high rotating speeds, which may lead to nonnegligible rotordynamic effects, including Coriolis force, spin softening, and stress stiffening effects. ese effects on the vibration of mistuned bladed disks are seldom discussed in available investigations. In this paper, the vibration characteristics of rotating mistuned bladed disks are addressed by considering these rotordynamic effects. First, finite element (FE) models of bladed disks are used to obtain the governing equations of motion, and an efficient method for getting the stress stiffening matrix of sector model is developed. en, the effective component-mode mistuning method (CMM) is employed to create compact, yet accurate, reduced-order models (ROMs). Finally, the models are validated and used to study the influences of Coriolis force, spin softening, and stress stiffening effects on the vibration of bladed disks with frequency mistuning factors. Numerical results show that these rotordynamic effects could significantly affect the vibrations of mistuning bladed disks, especially in the ranges of high speed, and should be carefully considered during analysis.

Bladed disks of engine rotors usually operate at harsh conditions of high rotating speeds, which may lead to nonnegligible rotordynamic effects, including Coriolis force, spin softening, and stress stiffening effects. These effects on the vibration of mistuned bladed disks are seldom discussed in available investigations. In this paper, the vibration characteristics of rotating mistuned bladed disks are addressed by considering these rotordynamic effects. First, finite element (FE) models of bladed disks are used to obtain the governing equations of motion, and an efficient method for getting the stress stiffening matrix of sector model is developed. Then, the effective component-mode mistuning method (CMM) is employed to create compact, yet accurate, reduced-order models (ROMs). Finally, the models are validated and used to study the influences of Coriolis force, spin softening, and stress stiffening effects on the vibration of bladed disks with frequency mistuning factors. Numeric...

Grosses tempêtes qui m'avez secouru, Beau soleil qui m'a contrecarré, Il y a haine en moi, forte et de date ancienne, Et pour la beauté on verra plus tard. Je ne suis, en effet, devenu dur que par lamelles ; Si l'on savait comme je suis resté moelleux au fond. Je suis gong et ouate et chant neigeux, Je le dis et j'en suis sûr."

A l’interface de l’automatique, de la mecanique et de l’acoustique musicale, le controle des instruments de musique s’emploie a developper des methodes permettant de controler, en temps reel, leur son acoustique. Les controleurs utilises dans ce domaine s’appuient sur des modeles lineaires ne prenant pas en compte les non-linearites presentes dans le comportement de certains instruments. Les gongs d’opera chinois presentent ainsi plusieurs phenomenes induits par des non-linearites geometriques, dont un tres caracteristique glissement frequentiel qui impacte plusieurs de leurs modes de vibration. Le present travail propose d’initier la mise en place d’un controle de ces instruments par le biais de trois etapes consecutives. Dans un premier temps, les performances et limites du controle modal moderne vis-a-vis des phenomenes non lineaires presents dans le comportement de l’instrument (distorsion harmonique, glissement frequentiel, resonances internes) sont etudiees et quantifiees. Les...

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Rotordynamics is a very challenging field because of machine complexities. Many internal and external factors contribute toward change in the structural dynamic characteristics. One of these factors is broad-band high-vibration amplitudes. In this article, a similar high vibration issue on a gas turbine is investigated using bode, orbit, and shaft centerline plots. Data from proximity probes installed on turbine generator system are captured and analyzed against any factor contributing toward high vibration issue. Fish bone diagram was used for root cause investigation. Main components investigated for the root cause of high vibration issue include generator rotor and casing. Rotor behavior has been examined by capturing orbit and shaft centerline diagrams, whereas casing contribution has been investigated by conducting operating deflection shape analysis. A comparison is drawn between a machine suffering from high vibration issue and a normal machine. Resonance was identified as th...

Dans les moteurs d'avion, l'utilisation de paliers inter-arbres est préconisée en vue d'un gain de masse non négligeable. Cependant, ce palier est très fortement sollicité par les excitations générées par les balourds présents sur les rotors et il est nécessaire d'introduire un amortissement supplémentaire pour préserver l'intégrité de la machine. La solution technologique retenue consiste à introduire un amortisseur à film fluide au niveau de ce palier inter-arbres. Cependant, ce type d'amortisseur peut engendrer des instabilités similaires à celles rencontrées dans les paliers hydrodynamiques classiques. On présente dans cet article un dispositif d'essais birotor permettant de tester différentes technologies de palier inter-arbres. Les essais montrent, pour des rotors co et contra rotatifs, que l'intégration d'un film fluide au niveau du palier inter-arbres permet d'atténuer les vibrations des rotors et confirment qu'il apparaît, dans certaines conditions, des instabilités.

A mesoscale turbocompressor spinning above 500,000 RPM is evolutionary technology for micro turbochargers, turbo blowers, turbo compressors, micro-gas turbines, auxiliary power units, etc for automotive, aerospace, and fuel cell industries. Objectives of this work are: 1) ...

Ce travail de recherche présente une technique numérique pour calculer la réponse vibratoire nonlinéaire d'une structure hyperélastique en contact avec un plan rigide. Les non-linéarités traitées sont similaires à celles prises en compte dans la simulation de la réponse dynamique des pneumatiques. La formulation mathématique de la méthode proposée est détaillée. La méthode est appliquée pour calculer les vibrations non-linéaires d'un cylindre rainuré en roulage sur un sol rigide. Les résultats de simulation sont analysés pour donner ensuite des éléments de validation de la stratégie numérique proposée. Mots clés-dynamique non linéaire, contact mécanique, modélisation multi-échelle

The study of rotating machines is usually carried out taking into account linearized hydrodynamic forces, considering dynamic coefficients of stiffness and damping, although a high order of nonlinearity can be significantly present in the system. To solve the nonlinear problem, the solution of Reynolds equation is practically mandatory for each time step in the numerical integration procedure, leading to high computational costs that often can make its application unavailable. In this paper, the validity of linear approximation for the oil film forces is discussed when the system operates under specific conditions, pointing out the influence of critical phenomena and dynamic parameters in rotordynamic analysis. Experimental tests are compared to numerical simulations for linear and nonlinear models of bearings in laboratory test rig in order to validate the analysis. Afterward, several simulations were accomplished, in time domain, for a rotor configuration more susceptible to critical operation, comparing the results for linear and nonlinear models. The main focus is on the influence of internal damping, gyroscopic effects, journal eccentricities, and excitation forces. The results demonstrate that the excitation force plays a fundamental role in nonlinearity degree of response, namely in extreme operation conditions

Turbogenerator as a set of turbines and electric generator presents a very complicated mechanical system. Its design must take into consideration the rotordynamics of rotor, which undergoes bending and torsional oscillation. The oscillation of the rotor depends upon its geometry and type of support. In this paper the dynamic models for the turbogenerator with three turbines and an electric generator supported in different types of bearings has been analysed. The mathematic model was extended with electromagnetic moment and the resistant moment of the working machine. The adopted models – mechanical and its mathematical - are a good base for dynamic analysis at the

Many rotating machines such as compressors, turbines, and pumps have long thin shafts with resulting vibration problems. They would benefit from additional damping near the center of the shaft. Magnetic dampers have the potential to be employed in these machines because they can operate in the working fluid environment, unlike conventional bearings. This paper describes an experimental test rig that was set up with a long thin shaft and several masses to represent a flexible shaft machine. An active magnetic damper was placed in three locations: near the midspan, near one end disk, and close to the bearing. With typical control parameter settings, the midspan location reduced the first mode vibration 82 percent, the disk location reduced it 75 percent, and the bearing location attained a 74 percent reduction. Magnetic damper stiffness and damping values used to obtain these reductions were only a few percent of the bearing stiffness and damping values. A theoretical model of both th...

The successful industrial application of flexible rotors supported on active magnetic bearings (AMBs) requires careful attention not only to rotordynamic design aspects, but also to electromagnetic and feedback control design aspects. This paper describes the design, construction and modeling process for an AMB test rig which contains a 1.23m long flexible steel rotor, with a mass of 44.9 kg and two gyroscopic disks. The rotor typifies a small industrial centrifugal compressor designed to operate above 12,000 rpm and the first bending natural frequency. There are four AMBs-two AMBs at the shaft ends to support the shaft with a combined load capacity of 2600N and two additional AMBs at the mid and quarter spans to allow for the application of simulated destabilizing fluid or electromagnetic forces to the rotor. Simulated aerodynamic cross coupling stiffness values are to be applied to the rotor through these two internal AMBs with the goal of developing stabilizing robust controllers. The unique design allows multiple support and disturbance locations providing the ability to represent a variety of machine configurations, e.g., between bearing and overhung designs. The shaft transfer function in lateral movement has been developed with finite element model and then verified by experimental frequency response measurements. Models for the power amplifiers, position sensors, signal conditioning and data converter hardware were developed, verified experimentally and included in the overall system model. A PID controller was developed and tuned to levitate the rotor and enable further system characterization.

Squeeze film dampers are widely used in aircraft engines, land-based gas turbines, and other rotating machines to improve system damping. They often have entrained air within the oil film, which is usually not taken into account in rotordynamics analysis due to lack of good formulations of the effects involved. The effects on the damping force calculation can be significant. This work presents a new formulation of the nonlinear Reynolds equation pressure evaluation within the squeeze film damper, including the effects of entrained air, with resulting changes in effective lubricant density and viscosity. Viscosity and density expressions are developed as a function of the air/oil volume fraction. The density of the bubbly oil is a function of the air bubble diameter, which changes due to surface tension effects during lubricant motion in the bubbly oil film. The lubricant viscosity decreases due to the entrained volume of air but increases due to the surface tension effects taken from experimental tests. Pressure supply and bubbly oil film cavitation effects are included in the analysis and end seal effects are evaluated. The nonlinear time-transient forces in the squeeze film damper are evaluated as functions of (1) lubricant and air properties; (2) damper geometry including diameter, length, clearance; (3) end seal properties; and (4) shaft position and velocity. Example cases of pressure calculations and radial and tangential forces are shown. Example nonlinear transient motions are presented for a rigid, symmetrical rotor and for a nonsymmetrical rotor representing a gas turbine-type fan rotor.

High Speed PM Motors are popularly used in Spacecraft flywheel application. So, this paper proposes two different designs: one for classical sizing and the other for genetic one at 50 kW output power with 250 m/s tip speed. These two design trials of High Speed Permanent Magnet Synchronous Motors (HSPMSMs) are simulated with MATLAB, and Genetic Algorithm Toolbox. Then, the two design examples comparisons are introduced in analytical forms with study of machines waveforms, Total Harmonic Distortion (THD) and rotor losses. Finally, the effect of poles number on some sizing parameters and motor characteristics are illustrated in the form of charts with the waveforms. This is done at constant number of slots and phases with number of poles are 4, 6, and 12 at 50 kW output power and tip speeds of 250, 200, 150, and 100m/s.

This research focuses on the structural analysis of aircraft electrical harnesses for the ± 270 VDC network. A combined experimental and simulation-based approach is used, where numerical models for the harnesses are developed and experimental data are acquired for basic test cases (steady load, pendulum test, etc.) as well as for different vibration load cases. Validation of the numerical models is done using the preliminary experimental data, resulting in a mature refined model in order to simulate complex scenarios. The goal of the project is thus to develop a suite of tools that allows detailed structural analysis of the aircraft harnesses, both stand alone as well as in combination with other systems.

Because of their high load carrying capacity and low cost, journal bearings are widely used to support the rotor of industrial machinery with high loads, such as steam turbines, centrifugal compressors and pumps. However, sudden catastrophic failure journal bearings can result in huge economic loss and high safety problems. It is necessary to develop effective condition monitoring technologies to detect and diagnose the failures at early stage and avoid such catastrophic failure. Previous researchers have studied the low frequency vibration characteristics as well as the high frequency vibration and acoustics emission for the early detection of journal bearing failure. However, these studies give relatively little information to the vibro-acoustic characteristics of high speed self aligning journal bearings. This paper focuses on the condition monitoring of high speed self aligning journal bearings through vibro-acoustic analysis.

Helicopter rotor track and balance (RTB) is the process of reducing to a minimum the rotor induced vibration present during helicopter flight. We propose here the use of an optimisation algorithm to reduce the number of dedicated flights required to perform RTB and return an aircraft to a serviceable state. The algorithm has recently been flight trialled by UK MoD on their Puma aircraft as part of a project to embody the algorithm on their existing RTB equipment. Both main and tail rotor applications were tested. Some of the results from that flight trial are shown here along with their effects on the development of the algorithm itself. The outcome indicates that there is significant potential cost saving and increased availability benefits to be made by using the algorithm across the fleet. Background The Minimum Flight Routine (MFR) algorithm was first conceived in early 2006, development then commenced in the summer of 2006 through a Knowledge Transfer Program (KTP) between Heli...

Y © J o h n W il e y & S o n s , I n c . C o l a b o r a c i ó n e n l a t r a d u c c i ó n : R a f a e l G a r c í a D í a z 7 ' I n g e n i e r o e n m i n a s p o r l a U n i v e r s i d a d d e G u a n a j u a t o , M é x i c o . R e v i s i ó n : M a r c e l o S a n t a l ó S o r s C a t e d r á t i c o d e m a t e m á t i c a s . La p r e s e n t a c i ó n y d is p o s ic i ó n e n c o n j u n t o d e s o n p r o p i e d a d d e l e d i t o r . N i n g u n a p a r t e d e e s t a o b r a p u e d e s e r r e p r o d u c i d a o t r a n s m i t i d a , m e d i a n t e n i n g ú n s i s t e m a o m é t o d o , e l e c t r ó n i c o o m e c á n ic o ( in c l u y e n d o e l f o t o c o p i a d o , l a g r a b a c i ó n o c u a l q u i e r s i s t e m a d e r e c u p e r a c i ó n y a l m a c e n a m i e n t o d e in f o r m a c i ó n ), s in c o n s e n t i m i e n t o p o r e s c r i t o d e l e d i t o r . D e r e c h o s r e s e r v a d o s : © 2 0 1 2 , EDITORIAL LIMUSA, S.A. d e C . V . GRUPO NOR IEG A EDITORES B a l d e r a s 95, M é x i c o ,

Le calcul numérique de fonctions de matrices f (A) avec A matrice carrée de Toeplitz ou Toeplitz-like de taille n × n trouve son intérêt dans divers domaines mathématiques, que ce soit pour la discrétisation d'une équation integro-différentielle partielle ou la construction de systèmes de filtres. Or, les méthodes classiques de calcul de fonctions de matrices n'utilisant aucune structure particulière de la matrice A, celles-ci sont alors de complexité O(n 3 ). Dans cette thèse nous cherchons à réduire cette complexité à O(n 2 ) voire O(n log 2 (n)) en exploitant la structure Toeplitz-like de A, notamment à l'aide de l'approximation rationnelle de notre fonction f . Après avoir donné quelques rappels concernant les fonctions de matrices et leur approximation, nous définissons une arithmétique sur les matrices Toeplitz-like, formant un sous-ensemble de matrices permettant des opérations rapides, comme l'addition, la multiplication ou l'inversion, réduisant le calcul d'une fonction rationnelle de matrice à une complexité O(n 2 ) voire O(n log 2 (n)). A l'aide de cette nouvelle arithmétique, nous nous attardons ensuite sur l'approximation des fonctions de matrice racine carrée et signe pour lesquelles nous reprenons et accélérons la méthode de Newton sous ses différentes formes. Enfin nous nous intéressons à l'approximation des fonctions de matrices f (A) par r(A) lorsque f est une fonction de Markov avec f : z → β α dµ(x) z-x où µ est une mesure positive à support dans [α; β] et A une matrice de Toeplitz symétrique. Nous énonçons alors l'un de nos principaux résultats qui est une borne supérieure pour l'erreur relative d'interpolation rationnelle 1 -r/f sur l'intervalle spectral de A. Cette borne est ensuite optimisée par un choix particulier des points d'interpolation. Nous discutons alors de la précision de trois représentations différentes de nos interpolants rationnels et appuyons nos résultats d'applications numériques pour un argument scalaire. Un résultat sur la positivité des paramètres de la fraction continue de Thiele dans le cas de fonctions de Markov est également démontré. Ces résultats sont enfin appliqués au cas matriciel et nous proposons diverses reformulations a priori et a posteriori de notre borne sur 1 -r/f pour une fonction de Markov f et un argument matriciel. De nombreuses expériences numériques illustrent notre démarche.

This project aims to design and create an affordable yet efficient small-scale wind turbine to provide people in diverse environments with the opportunity to harness wind energy. We investigated different wind turbine designs and ultimately decided on a design with a three-bladed horizontal axis wind turbine placed inside of a converging-diverging nozzle, in order to maximize wind speeds, and furthermore, maximize power output. We also made our design accessible to others by 3D printing our wind turbine, since 3D printing is a fast growing and accessible technology. As a result, we used and investigated ABS plastic because it is one of the most common materials used in 3D printers. Additionally, we used PVC piping for the tower supporting our nozzle and we constructed a platform out of wood. Lastly, we used fiberglass coated with a two-part epoxy to construct the convergingdiverging nozzle. Our initial wind turbine design was tested without the converging-diverging nozzle. We conducted indoors tests with high-speeds fans producing wind speeds ranging between 3.5 m/s to 13.5 m/s, however, the turbine did not rotate. Due to major modifications that needed to be made to our original design in order to create a working turbine, we were forced to redesign and reprint our wind turbine. This redesigned wind turbine was able to rotate without the converging-diverging nozzle at wind speeds of 4.8 m/s. When tested inside of the nozzle, it was determined that a minimum wind speed of 4.7 m/s was needed by the turbine to self-start within the converging-diverging nozzle. From the tests conducted on our redesigned turbine, we found that our wind turbine was able to produce 1 Watt of power. This project would not have been possible if it weren't for the help of our advisor Professor Rahbar who helped guide us in the correct direction with our goals for this project. Professor Rahbar was always quick to provide solutions to problems that arose especially those that involved material selection. Through his experience and knowledge of materials, Professor Rahbar has helped our team determine alternative materials for our wind turbine when one material proved to be unsuitable. Professor Rahbar also worked hard to make sure our team remained on track by always asking for ways he could help. We would also like to thank Professor Mathisen for helping us with our fluid analysis calculations and for helping us understand the complex fluid phenomena occurring within our converging-diverging nozzle as well as around our wind turbine. We would like to thank William (Bill) Grudzinski for his help in reviewing past year's wind reports and for suggesting various ideal locations to measure wind speeds. We would like to thank Russell Lang for his dedication to helping with the assembly of our project. For the toughest of our manufacturing dilemmas, he was always able to provide advice on how to best construct our project's components. He has helped us by allowing us to use the tools in his shop to build the mount and tower for our turbine as well as the foundation. He has also helped us by suggesting methods for constructing our converging-diverging nozzle and then ultimately putting us in contact with Matt Wilkinson from Polycell to construct the foam mold for the nozzle. With that, we would like to thank Mr. Wilkinson and all those at Polycell who helped in the creation of our C-D nozzle. We would like to thank Professor John Orr and the WPI Task Force on Sustainability for agreeing to fund us past the total funding the Mechanical Engineering Department provides ME students. This help has allowed us to fully test our theories and understand the budgeting challenges many engineers face. Additionally, we would like to thank WPI's Mechanical Engineering Department for the financial help that funded the majority of our project. More specifically, we would like to thank Barbara Furham for walking our team through the setup of our budget through purchasing the materials for our project. We would like to thank Erica Stults for 3D printing the components to our turbine. We would also like to thank Professor Scarpino and Peter Hefti. Each has helped us with the understanding and testing of our first generator. Last but not least, we would like to thank William (Bill) Appleyard in the Electrical and Computer Engineering department for providing us with a smaller and much more effective generator motor. He also provided us with numerous manufacturing and electrical solutions to problems that arose. In conclusion, our team is thankful for all the help we have received from those mentioned above.

Rotating disk and blade fatigue failures are usually a low percentage of failures in most machinery types, but other than coupling / shaft end failures remain some of the most problematic for extensive repairs. High-cycle fatigue failures of rotating disks and blades are not common in most machinery types, but when they occur, they require extensive repairs and resolution can be problematic. This paper is an update of the tutorial given at the 2004 Turbomachinery Symposium focusing on high-cycle fatigue failures in steam turbines, centrifugal and axial gas compressors in refineries and process plants. The failure theories and many of the descriptions for cases given in 2004 have been updated to include blade resonance concerns for potential flow as well as vane and blade wake effects. Disk vibratory modes can be of concern in many machines, but of little concern in others as will be explained. In addition, vibratory modes are included where blades are coupled via communication with the main disk. Over the past decade, fluid-structure-interaction computational methods and modal testing have improved and have been applied to failure theories and problem resolution in the given cases. There is also added information on the effects of mistuning blades and disks, some beneficial and some with serious concerns for increased resonant amplification. Finally, knowledge about acoustic pressure pulsation excitation, particularly for centrifugal impellers at rotating blade passing frequency, has been greatly expanded. A review of acoustics calculations for failure prevention, mainly for high-pressure applications is covered here.

A machine-tool structure optimization is an important technique that improves the machining efficiency and saves materials and the energy resource. In this work, dynamic design optimization method for Machine-Motorized-Spindle (MMS) subjected to a number of rotating unbalanced forces effects is presented. Linear forced-Rotordynamic analysis with design explorer method has been used to simulate the output response. The Design Variables (DVs) and their limits were carefully chosen and applied to develop the Design-of-Experiment (DOE). The Box-Behnken Design (BBD) method, because of its good organization in providing much information in a minor number of required statistical experiments was used to generate the DOE. The influences of DVs on the dynamic of MMS and their levels optimization were evaluated by utilizing the Response-Surface (RS) method. The results showed that the spindle shaft inner diameter of the motor-rotor seat and its rotating unbalanced mass, and modulus of elastici...

Electromagnetic actuators that use a current-carrying coil (which is placed in a magnetic field) to generate mechanical force are conceptually attractive components for active control of rotating shafts. In one concept that is being tested in the laboratory, the control forces from such actuators are applied on the flexibly supported bearing housings of the rotor. Development of this concept into

One of the causes of rotor instability is the damping of the rotating parts of the system. Its destabilizing effect can be counteracted only by introducing a suitable non rotating damping, with the instability threshold increasing with increasing ratio between non rotating and rotating damping. The aim of the present paper is showing that support anisotropy tends to increase the stability of the system, an effect here shown to take place in both cases of viscous and hysteretic damping. A heuristic explanation of the stability increase due to support anisotropy can be given considering that anisotropic supports usually cause the orbits to be elliptical, which increases energy dissipation due to rotating damping. Actually, no increase of the instability threshold was observed in a case in which the orbits remain circular in spite of the anisotropy of the supports.

Centrifugal softening effect is an alleged and elusive reduction of the natural frequencies of a rotating system with increasing speed sometimes found in finite element rotordynamics. This reduction may, in some instances, be large enough to cause some of the natural frequencies to vanish, leading to a sort of elastic instability. Some doubts can be however cast on the phenomenon in itself and on the mathematical models causing it to appear. The aim of the present work is to shed some light on centrifugal softening and to discuss the assumptions that are at the basis of 3-dimensional finite element (FEM) modeling in rotordynamics.

Nonsynchronous rotating damping, i.e. energy dissipations occurring in elements rotating at a speed different from the spin speed of a rotor, can have substantial effects on the dynamic behaviour and above all on the stability of rotating systems. The free whirling and unbalance response for systems with nonsynchronous damping are studied using Jeffcott rotor model. The system parameters affecting stability are identified and the threshold of instability is computed. A general model for a multi-degrees of freedom model for a general isotropic machine is then presented. The possibility of synthesizing nonsynchronous rotating and nonrotating damping using rotorand stator-fixed active dampers is then discussed for the general case of rotors with many degrees of freedom.

DYNROT is a code based on the finite element method aimed to perform a complete study of the dynamic behaviour of rotors. Although initially designed to solve the basic linear rotordynamic problems (Campbell diagram for damped or undamped systems, unbalance response, critical speeds, static loading), it can be used for the study of nonstationary motions of nonlinear rotating systems and for the torsional analysis of rotors and reciprocating machines. 
One of the distinctive features of the code is the use of complex co-ordinates, both for isotropic and nonsymmetric systems. Extensive use of complex arithmetics is then made in all parts of the analysis. The modal approach is used in some of the solution routines to increase the efficiency of the computation or to compute an equivalent viscous damping in the cases where hysteretic damping cannot be entered directly into the model.

En este trabajo se presenta el desarrollo del modelo matemático de un identificador algebraíco en línea; para determinar el desbalance y su posición angular en sistemas rotodinámicos vibratorios de múltiples grados de libertad. El identificador propuesto se obtiene al tomar como base un modelo matemático en elemento finito para sistemas rotatorios de múltiples grados de libertad. Para el desarrollo del modelo, se consideró un elemento tipo viga con cuatro grados de libertad por nodo, donde se incluyen los efectos de la inercia rotatoria, momentos giroscópicos, deformaciones por cortante y amortiguamiento interno y externo del sistema. Se evaluó y analizó el comportamiento en el tiempo del identificador propuesto, para la identificación del desbalance y su posición angular; se tomó como dato de entrada la respuesta de vibración al desbalance a velocidad constante del sistema rotodinámico, obtenida de la simulación numérica.

This paper evaluates how tilting pad bearings can affect the forced response of a vertical rotor. The stiffness and damping for a tilting pad bearing is dependent on the position in the bearing (load on pad or load between pad) and the magnitude of the load. In a vertical machine the position of the shaft in the bearing will change for each time-step and produce a periodic stiffness and damping. This periodic stiffness has shown to produce higher frequency components depending on the number of pads. For the four pad bearing considered in this paper these frequency components are 3Ω and 5Ω in the stationary coordinate system. The aim of this study is to evaluate if the periodic coefficients in the bearing could excite the system and cause resonance problems. It is found that for high loads the system can be excited due to the bearing dynamics at Ω/ω n = 0.33 and Ω/ω n = 0.2. However at low loads this effect is negligible.