Hydrodynamic flexure pivot tilting pad gas bearings (FPTPGBs) can enable successful operation of oil-free microturbomachinery. This work presents the experimental and analytical study of such bearings with different damper configurations. A test rig was constructed that could safely operate a ~28.6 mm, 0.8 kg rotor beyond 120 krpm. A time domain orbit simulation, which integrates nonlinear equations of motion for the rotor-bearing elements, was implemented as the primary analysis tool to predict rotor-bearing responses to imbalance, the presence and location of critical speeds, etc. Complementary analyses were also performed with a model that uses linear bearing impedance coefficients to predict system natural frequencies. Imbalance response testing verified that the rotor-bearing system behaved linearly in the region above the critical speed, and orbit simulations predicted the response to a calibrated imbalance with notable agreement. Viscoelastic dampers added behind the FPTPGB pads delayed the onset of subsynchronous vibrations (from 43 krpm without damper to above 50 krpm with damper) of the system with bearing clearance increased by shims. Midrange subsynchronous vibrations initiated at ~20 krpm were eventually suppressed by ~25 krpm due to the stabilizing effect of rotor centrifugal growth. The viscoelastic dampers had a negligible effect on suppressing these midrange subsynchronous vibrations in experiments, but this was not demonstrated in simulations, presumably due to much lower stiffness contribution of the damper at lower frequencies. The ideal, perfectly aligned models in the simulations were able to tolerate shims up to only 10% of nominal clearance, but the test rig exhibited surprising stability with shims as much as 200% of nominal clearance; this increase may be caused by imposed eccentricities due to misalignments in the test rig. FPTPGBs supported by compliant bump foils can have the ability to tolerate rotor misalignments and shock loading like foil gas bearings. Simulation studies on imbalance response characteristics for several bearing shell mass and support stiffness configurations present initial design guidelines for the application. Namely, results showed that FPTPGBs favored large bearing shell mass and large support stiffness, while FPTPGBs with radial compliance favored small bearing shell mass with large support stiffness.

We de ne the quantum cluster character assigning an element of a quantum torus to each representation of a valued quiver (Q; d) and investigate its relationship to external and internal mutations of a quantum cluster algebra associated to (Q; d). We will see that the external mutations are related to re ection functors and internal mutations are related to tilting theory. Our main result will show the quantum cluster character gives a cluster monomial in this quantum cluster algebra whenever the representation is rigid, moreover we will see that each non-initial cluster variable can be obtained in this way from the quantum cluster character.

Given a quiver flag variety Y equipped with a tilting bundle E, a construction ofCraw, Ito and Karmazyn [CIK18] produces a closed immersion f_E : Y -> M(E), where M(E) is the fine moduli space of cyclic modules over the algebra End(E).In this thesis we present two classes of examples where f_E is an isomorphism. Firstly, when Y is toric and E is the tilting bundle from [Cra11]; this generalises the well-known fact that P^n can be recovered from the endomorphism algebra of \oplus_{0\leq i \leq n} O_{P^n}(i). Secondly, when Y = Gr(n, 2), the Grassmannian of 2-dimensional quotients of k^n and E is the tilting bundle from [Kap84]. In each case, we give a presentation of the tilting algebra A = End(E) by constructing a quiver Q' such that there is a surjective k-algebra homomorphism \Phi: kQ' -> A, and then give an explicit description of the kernel.

This thesis provides experimental frequency dependent stiffness and damping coefficient results for a high-speed, lightly loaded, flexible-pivot tilting-pad bearing, with a load-on-pad configuration. Test conditions include four shaft speeds (6000, 9000, 13000 and 16000 rpm), and bearing unit loads from 172 kPa to 690 kPa. The results show that the bearing stiffness is a quadratic function of the frequency of vibration; hence their frequency dependency can be modeled by added-mass terms. The additional degrees of freedom introduced by the pads and the influence of the inertial forces generated in the fluid film account for this frequency dependency. The conventional frequency-dependent stiffness and damping model for tilting-pad bearings is extended with an added-mass matrix to account for the frequency dependency. This approach allows the description of the bearing dynamic characteristics with frequency-independent stiffness, damping and added-mass matrices. Experimental results are compared with predictions from the Reynolds equation and from a bulk-flow Navier-Stokes model. Both models produce good predictions of the stiffness and damping coefficients. However, results show that the bulk-flow model is more adequate for predicting the direct added-mass terms because it accounts for the fluid inertial forces. A bulk-flow solution of the Navier-Stokes equations that includes the effects of fluid inertia should be used to calculate the rotordynamic coefficients of a flexible-pivot tilting-bearing. Static performance measurement results are also detailed. Results include pad metal temperatures, eccentricity-ratios and attitude-angle as a function of bearing load, and estimated power losses.

Random variables defined on the natural numbers may often be approximated by Poisson variables. Just as normal approximations may be improved by saddlepoint methods, Poisson approximations may be substantially improved by tilting, expansion, and other related methods. This work will develop and examine the use of these methods, as well as present examples where such methods may be needed. / Ph. D.

In the theoretical analysis of high speed rotor bearing systems, it is common to use four displacement and four velocity based coefficients, which characterise the behaviour of the lubricating fluid film. Although a great deal of work has been published establishing theoretical models of all types of hydrodynamic journal bearings, the large amount of experimental work has centred on relatively low speed conditions. This work presents a contribution to the experimental study of the static and dynamic characteristics of oil films in journal bearings used in high-speed rotating machinery. The main objectives of the work are: • To devise new experimental techniques for the measurement of dynamic coefficients suitable for use at high rotational speeds • To design, manufacture, assemble and commission a test facility to measure the static and dynamic characteristics of journal bearings at speeds up to 30000 rpm • To determine the static and dynamic characteristics of a 5 Pad Tilting Pad Journal Bearing Unit of 80 mm diameter at speeds up to 25 000 rpm using the said test facility. New techniques are particularly necessary for the measurement of velocity coefficients because these invoke the necessity of imposing a velocity on to the bearing housing and previous techniques have utilised synchronous motion of the bearing. Consequently a new experimental procedure for measuring the four velocity or damping coefficients of an oil film journal bearing from imposed dynamic "orbits" has been devised called the "double pulse" technique. All four velocity coefficients are derived from one imposed journal centre dynamic orbit and, therefore may be regarded as being obtained at the same time. The method requires the production of a "cross- over" point similar to that of a "figure of eight" shaped orbit and utilises the "cross-over" point therein. Coefficients are initially evaluated in a co-ordinate system, which is chosen to align with the designated parts of the measured orbit. Each coefficient is then evaluated from single values of instantaneous imposed force and resulting journal centre velocity. Coefficients are them converted into any other desired axes system. The result is a simpler experimental procedure, with reduced uncertainty compared to hitherto existing methods. The use of non-sinusoidal excitation of the oil film was explored, in the form of applying a step-pulse train load pattern to produce a cross-over pattern in the journal displacement ·orbit'. Experimental tests were completed on a tilting pad bearing at speeds up to 15 000 rpm inclusive. At speeds above this, the bearing exhibited a vibrational response, which precluded the accurate measurement of journal centre displacement.

A novel tilting pad journal bearing model including pivot flexibility as well as temporal fluid inertia effects on the thin film fluid flow aims to accurately predict the bearing forced performance. The predictive model also accounts for the thermal energy transport effects in a TPJB. A Fortran program with an Excel GUI models TPJBs and delivers predictions of the bearing static and dynamic forced performance. The calculation algorithm uses a Newton-Raphson procedure for successful iterations on the equilibrium pad radial and transverse displacements and journal center displacements, even for bearings pads with very soft pivots. The predictive model accounts for the effect of film temperature on the operating bearing and pad clearances by calculating the thermal expansion of the journal and pad surfaces. The pad inlet thermal mixing coefficient (lambda) influences moderately the predicted fluid film temperature field. Pad pivot flexibility decreases significantly and dominates the bearing stiffness and damping coefficients when the pivot stiffness is lower than 10% of the fluid film stiffness coefficients (with rigid pivots). Pivot flexibility has a more pronounced effect on reducing the bearing damping coefficients than the stiffness coefficients. Pad pivot flexibility may still affect the bearing behavior at a light load condition for a bearing with a large pad preload. Pad pivot flexibility, as well as the fluid inertia and the pads’ mass and mass moment of inertia, could influence the bearing impedance coefficients, in particular at high whirl frequencies. The stiffness and damping coefficients of a TPJB increase with a reduction in the operating bearing and pad clearances. The work delivers a predictive tool benchmarked against a number of experimental results for test bearings available in the recent literature. The static and dynamic forced performance characteristics of actual TPJBs can not be accurately predicted unless their pad flexibility and pivot flexibility, fluid film temperature, pad inlet thermal mixing coefficient, operating bearing and pad clearances, among others are well known in advance. However, the extensive archival literature showcasing test procedures and experimental results for TPJBs does not report the above parameters. Thus, reasonable assumptions on the magnitude of certain elusive parameters for use in the predictive TPJB model are necessary.

Submitted by Marco Antônio de Ramos Chagas (mchagas@ufv.br) on 2015-11-12T13:17:51Z No. of bitstreams: 1 texto completo.pdf: 1082850 bytes, checksum: e652565e5953a1e93915f35cfdcaf7f4 (MD5) / Made available in DSpace on 2015-11-12T13:17:51Z (GMT). No. of bitstreams: 1 texto completo.pdf: 1082850 bytes, checksum: e652565e5953a1e93915f35cfdcaf7f4 (MD5) Previous issue date: 2015-01-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho apresentamos as categorias cluster, que foram introduzidas por Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten e Gordana Todorov, com o objetivo de categoriíicar as algebras cluster criadas em 2002 por Sergey Fomin e Andrei Zelevinsky. Os autores acima, em [4], mostraram que existe uma estreita relação entre algebras cluster e categorias cluster para quivers cujo grafo subjacente é um diagrama de Dynkin. Para isto desenvolveram uma teoria tilting na estrutura triangulada das categorias cluster. Este resultado foi generalizado mais tarde por Philippe Caldero e Bernhard Keller em [8] para quivers do tipo acíclico. O objetivo principal desta dissertação e estudar como a teoria tilting sobre cluster permite estabelecer a relação entre estas estruturas e apresentar exemplos. / In this work we present the cluster categories, which were introduced by Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten and Gordana Todorov, with objective of categoriíication cluster algebras created in 2002 by Sergey Fornin and Andrei Zelevinsky. The authors above, on [4], showed that there is a close relationship between cluster algebras and cluster categories for quivers whose un- derlying graph is a Dynkin diagrarn. For this they develOped a tilting theory in the triangulated structure of the cluster categories. This result was later generalized by Philippe Caldero and Bernhard Keller on [8] for quivers of the acyclic type. The main objective of this dissertation is to study how the tilting theory about cluster enables establish the relationship between these structures and present examples.

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