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This article presents a methodology for calculating the gains of an output feedback controller for active vibration control of flexible rotors. The methodology is based on modal reduction. The proportional and derivative gains are obtained by adjusting the first two damping factors of the system and keeping the lengths of the two eigenvalues constant in the real-imaginary plane. The methodology is applied to an industrial gas compressor supported by active tilting-pad journal bearings. The unbalance response functions and mode shapes of the flexible rotor with and without active control are presented, showing significative improvement in damping reserve with the control. The importance of sensor location is emphasized, on the basis of the energy necessary to operate the active system over the entire frequency range studied. The best results are obtained by a decentralized controller, observing displacement and velocity of the shaft at the bearing positions.

This is Part I of a two-part series of papers describing the effects of high-pressure injection pockets on the operating conditions of tilting-pad thrust bearings. In Part I a numerical model based on the Reynolds equation is developed extending the three-dimensional thermoelastohydrodynamic (TEHD) analysis of tilting-pad thrust bearings to include the effects of high-pressure injection and recesses in the bearing pads. The model is applied to the analysis of an existing bearing of large dimensions and the influence of the pocket is analyzed. In the analysis, the high-pressure oil injection used for hydrostatic jacking is turned off (i.e., only the effect of the pocket is studied). It is shown that a shallow pocket positively influences the performance of the bearing because it has characteristics similar to those of a Rayleigh-step bearing. In Part II of the paper (Heinrichson, N., Fuerst, A., and Santos, I. F., 2007, ASME J. Tribol., 129(4), pp. 904–912) measurements of pressure profiles and oil film thickness for a test-pad are compared to theoretical results. The analysis of Part II deals both with flow situations, where the high-pressure injection is turned off, as well as with situations where it is turned on for hydrostatic jacking.

This is Part II of a two-part series of papers describing the effects of high-pressure injection pockets on the operating conditions of tilting-pad thrust bearings. The paper has two main objectives. One is an experimental investigation of the influence of an oil injection pocket on the pressure distribution and oil film thickness. Two situations are analyzed: (i) when the high-pressure oil injection is turned off and (ii) when the high-pressure injection is turned on. The other objective is to validate a numerical model with respect to its ability to predict the influence of such a pocket (with and without oil injection) on the pressure distribution and oil film thickness. Measurements of the distribution of pressure and oil film thickness are presented for tilting-pad thrust bearing pads of ∼100cm2 surface area. Two pads are measured in a laboratory test rig at loads of ∼1.5MPa and ∼4.0MPa and velocities of up to 33m∕s. One pad has a plain surface. The other pad has a conical injection pocket at the pivot point and a leading-edge taper. The measurements are compared to theoretical values obtained using a three-dimensional thermoelastohydrodynamic (TEHD) numerical model. At the low load, the theoretical pressure distribution corresponds well with the measured values for both pads, although the influence of the pocket is slightly underestimated. At the high load, large discrepancies exist for the pad with an injection pocket. It is argued that the discrepancies are due mainly to geometric inaccuracies of the collar surface, although they may to some extent be due to the simplifications employed in a Reynolds equation description of the pocket flow. The measured and theoretical values of oil film thickness compare well at low loads and velocities. At high loads and velocities, discrepancies grow to up to 25%. This is due to the accuracy of the measurements. When using hydrostatic jacking the model predicts the start-up behavior well.

The highest vibration levels in large pump turbines are, in general, originated in the rotor stator interaction (RSI). This vibration has specific characteristics that can be clearly observed in the frequency domain: harmonics of the moving blade passing frequency and a particular relationship among their amplitudes. It is valuable for the design and condition monitoring to count on these characteristics. A CFD model is an appropriate tool to determine the force and its characteristics. However, it is time consuming and needs highly qualified human resources while usually these results are needed immediately and in situ. Then, it is useful to determine these characteristics in a simple, quick, and accurate method. At present, the most suitable method indicates a large amount of possible harmonics to appear, without indicating the relative importance of them. This paper carries out a theoretical analysis to predict and explain in a qualitative way these frequencies and amplitudes. The theoretical analysis incorporates the number of blades, the number of guide vanes, the RSI nonuniform fluid force, and the sequence of interaction. This analysis is compared with the method currently in use, and both methods are applied to a practical case. The theoretical analysis gives a resulting force over the pump turbine, which corresponds well to the measured behavior of a pump turbine in terms of its frequencies and the relationship between their amplitudes. A corrective action is proposed as a result of the analysis and after it is carried out in one of the units, the vibration levels are reduced. The vibration induced by the RSI is predicted considering the sequence of interaction and different amplitudes in the interactions between the same moving blade and different stationary blades, giving a different and original interpretation about the source of the vibration characteristics. A successful corrective action is proposed as a consequence of this new interpretation.

Noise contaminated zero problems involve functions that cannot be evaluated directly, but only indirectly via observations. In addition, such observations are affected by a non-deterministic observation error (noise). We investigate the application of numerical bifurcation analysis for studying the solution set of such noise contaminated zero problems, which is highly relevant in the context of equation-free analysis (coarse grained analysis) and bifurcation analysis in experiments, and develop specialized algorithms to address challenges that arise due to the presence of noise. As a working example, we demonstrate and test our algorithms on a mechanical nonlinear oscillator experiment using control based continuation, which we used as a main application and test case for development of the Coco compatible Matlab toolbox Continex that implements our algorithms.

tor-bearing systems have critical speeds and to pass through them is an ongoing challenge in the field of mechanical engineering. The incorporation of shape memory alloys in rotating systems has an increasing importance to improve system performance and to avoid potential damaging situations when passing through critical speeds. In this work, the feasibility of applying shape memory alloys to a rotating system is experimentally investigated. Shape memory alloys can change their stiffness with temperature variations and thus they may change system dynamics. Shape memory alloys also exhibit hysteretic stress-strain relations which may be utilized for damping purposes. These ideas are tested in this study on a dedicated test-rig, consisting of a rigid shaft and disc held vertically by passive magnetic bearings, where the damping is low. The bearing housings is flexibly supported by shape memory alloy helical springs, and because of high dynamic coupling between shaft and bearing housing, the shape memory alloy springs are able to reduce vibration in the shaft. The shape memory alloy springs are characterized by force-displacement tests in different temperatures. Transients of step perturbations and mass imbalance responses of the rotor-bearing system at different temperatures and excitation frequencies are carried out to determine the dynamic behaviour of the system. The behaviour and the performance in terms of vibration reduction and system adaptability are compared against a benchmark configuration comprised by the same system having steel springs instead of shape memory alloy springs. The experimental results clearly show that the stiffness changes and hysteretic behaviour of the shape memory alloys springs alter system dynamics both in terms of critical speeds and mode shapes. Vibration peaks could be reduced up to 47 % during ramp-up tests compared to the system configuration with steel springs and the two first critical frequencies could be altered up to 7 % by temperature changes.

Gas bearings are popular for their high speed capabilities, low friction and clean operation, but suffer from poor damping, which poses challenges for safe operation in presence of disturbances. Enhanced damping can be achieved through active lubrication techniques using feedback control laws. Such control design requires models with low complexity, able to describe the dominant dynamics from actuator input to sensor output over the relevant range of operation. The mathematical models based on first principles are not easy to obtain, and in many cases, they cannot be directly used for control design due to their complexity and parameter uncertainties. As an alternative, this paper presents an experimental technique for ”in situ” identification of low complexity models of the entire rotor-bearingactuator system. Using grey-box identification techniques, the approach is shown to be easily applied to industrial rotating machinery with gas bearings and to allow for subsequent control design. The paper shows how piezoelectric actuators in a gas bearing are efficiently used to perturb the gas film for identification over relevant ranges of rotational speed and gas injection pressure. Parameter-varying linear models are found to capture the dominant dynamics of the system over the range of operation. Based on the identified models, decentralised proportional control is designed and is shown to obtain the required damping in theory as well as in a laboratory test rig.

In this paper, a method for identifying uncertain parameters in a rotordynamic system composed of a flexible rotating shaft, rigid discs and two radial active magnetic bearings is presented. Shaft and disc dynamics are mathematically described using a Finite Element (FE) model while magnetic bearing forces are represented by linear springs with negative stiffness. Bearing negative stiffness produces an unstable rotordynamic system, demanding implementation of feedback control to stabilize the rotordynamic system. Thus, to identify the system parameters, closed-loop system identification techniques are required., The main focus of the paper relies on how to effectively identify uncertain parameters, such as stiffness and damping force coefficients of bearings and seals in rotordynamic systems. Dynamic condensation method, i.e. pseudo-modal reduction, is used to obtain a reduced order model for model-based control design and fast identification., The paper elucidates how nodal parametric uncertainties, which are easily represented in the full FE coordinate system, can be represented in the new coordinate system of the reduced model. The uncertainty is described as a single column vector of the system matrix A of the full FE model while it is represented as several elements spread over multiple rows and columns of the system matrix of the reduced model. The parametric uncertainty, for both the full and reduced FE model, is represented using Linear Fractional Transformation (LFT). In this way the LFT matrices represent the mapping of the uncertainties in and out of the full and reduced FE system matrices. Scaling the LFT matrices easily leads to the amplitudes of the uncertainty parameters., Youla Parametrization method is applied to transform the identification problem into an open-loop stable problem, which can be solved using standard optimization methods., An example shows how to decouple and identify an uncertainty in the linear bearing stiffness of a reduced FE rotordynamic system.

Fluid film forces are generated in multirecess journal bearings by two types of lubrication mechanism: the hydrostatic lubrication in the bearing recesses and hydrodynamic lubrication in the bearing lands, when operating in rotation. The combination of both lubrication mechanism leads to hybrid journal bearings (HJB). When part of hydrostatic pressure is also dynamically modified by means of hydraulic control systems, one refers to the active lubrication, resulting in the active hybrid journal bearing (AHJB). The AHJB is mathematically modeled based on the Reynolds' equations in land regions and a set of modified continuity equations for the lubricant flow in the bearing recesses, coupled to equations describing pressure and flow in the servovalves. The solution of such a set o equations allows the determination of fluid film stiffness and damping coefficients of the hybrid and/or active lubricated bearing. Such coefficients are a function of design and operational parameters, characterized by the Sommerfeld number as well by the gains of the feedback control system. The main contribution of the present theoretical work is to analyze the stability characteristics of a flexible rotor-bearing system by using passively and actively lubricated hybrid journal bearing. The dynamic of a flexible rotor is modeled by using finite element method and, after coupling the bearing dynamic coefficients to the rotor model, the feedback control law is defined and a suitable set of control gains is calculated for the active lubrication, leading to a rotor with a wider and more safety operational frequency range.

Parameter identification procedures and model validation are major steps towards intelligent machines supported by active magnetic bearings (AMB). The ability of measuring the electromagnetic bearing forces, or deriving them from measuring the magnetic flux, strongly contributes to the model validation and leads to novel approaches in identifying crucial rotor parameters. This is the main focus of this paper, where an intelligent AMB is being developed with the aim of aiding the accurate identification of damping and stiffness coefficients of active lubricated journal bearings. The main contribution of the work is the characterization of magnetic forces by using two different experimental approaches. Such approaches are investigated and described in detail. A special test rig is designed where the 4 pole - AMB is able to generate forces up to 1900 N. The high precision characterization of the magnetic forces is conducted using different experimental tests: (a) by using hall sensors mounted directly on the poles (precise measurements of the magnetic flux) and by an auxiliary system, composed of strain gages and flexible beams attached to the rotor, (b) by measuring the input current and bearing gap variations, monitoring the bearing input signals. Advantages and drawbacks of the different methodologies are critically discussed. The linearity ranges are experimentally determined and the characterization of magnetic forces with a high accuracy of less than 1% is achieved. (%error is normalized with respect to the instantaneous measured force obtained from the strain gauges signals).Copyright © 2006 by ASME

When implementing active control into bladed rotating machines aiming at reducing blade vibrations, it can be shown that blade as well as rotor vibrations can in fact be controlled by the use of only shaft-based actuation. Thus the blades have to be deliberately mistuned. This paper investigates the dynamical characteristics of a mistuned bladed rotor and shows how, why and when a bladed rotor becomes controllable and observable if properly mistuned. As part of such investigation modal controllability and observability of a tuned as well as a mistuned coupled rotor-blade system are analysed. The dependency of the controllability and observability on varying rotational speed and mode shape interaction phenomena between parametric and basis mode shape components are also analysed. Numerical results reveal a limitation of the achievable controllability and observability, once quantitative measures of modal controllability and observability converge toward steady levels as the degree of mistuning is increased. Finally, experimental control results are presented to prove the theoretical conclusions and to show the feasibility of controlling rotor and blade vibrations by means of shaft-based actuation in practice.

An active control system is developed to control blade as well as rotor vibrations in a coupled rotor–blade system where rotor lateral and blade flexible movements are coupled. In order to cope with the periodic time-varying dynamics of such systems, a periodic modal controller, based on time-variant modal analysis, is designed. The periodic time-varying equations of motion are transformed into a time-invariant modal form, which is used for designing the periodic controller. The modal decoupling of the equations of motion allows a system order reduction resulting in a simpler and more implementable controller addressing only specific vibration modes, which can cause problems. Moreover, the time-varying modal matrices, used for the modal decoupling, are also used for controllability and observability analyses in order to achieve optimal actuator and sensor placement in the system. The applicability of the controller design methodology is evaluated by a numerical example where a coupled periodic time-variant system built by a rigid rotor with four flexible blades is simulated. In the simulation model tip masses are added at the end of the blades with the aim of emphasizing the blade inertia effects and the time periodicity of the system. Rotor angular movements and the gyroscopic effect are neglected for simplicity, and the blades are modelled as flexible Bernoulli beams. Three different control schemes are designed using the described methodology. The results demonstrate that the designed controllers are capable to cope with the time periodicity of the system and suppress very efficiently only the vibration modes addressed.

In this paper the complete set of modified Reynolds’ equations for the active lubrication is presented. The solution of such a set of equations allows the determination of stiffness and damping coefficients of actively lubricated bearings. These coefficients are not just dependent on Sommerfeld number, as it would be the case of conventional hydrodynamic bearings, but they are also dependent on the excitation frequencies and gains of the control loop. Stiffness as well as damping coefficients can be strongly influenced by the choice of the control strategy, servo valve dynamics and geometry of the orifices distributed over the sliding surface. The dynamic coefficients of tilting-pad bearings with and without active lubrication and their influence on an industrial compressor of 391 Kg, which operates with a maximum speed of 10,200 rpm, are analyzed. In the original compressor design, the bearing housings are mounted on squeeze-film dampers in order to ensure reasonable stability margins during full load condition (high maximum continuous speed). Instead of having a combination of tilting-pad bearings and squeeze-film dampers, another design solution is proposed and theoretically investigated in the present paper, i.e. using actively lubricated bearings. By choosing a suitable set of control gains, it is possible not only to increase the stability of the rotor-bearing system, but also enlarge its operational frequency range.Copyright © 2003 by ASME

This work makes a theoretical and experimental contribution to the problem of rotor–blades dynamic interaction. A validation procedure of mathematical models is carried out with the help of a simple test rig, built by a mass–spring system attached to four flexible rotating blades. With this test rig, it is possible to highlight some dynamic effects and experimentally simulate the structural behaviour of a windmill in two dimensions (2-D model). Only lateral displacement of the rotor in the horizontal direction is taken into account. Gyroscopic effect due to angular vibrations of the rotor is eliminated in the test rig by attaching the rigid rotor to a flexible foundation. The blades are modelled as Euler–Bernoulli beams. Using three different approaches to describe the beam deformation one achieves: (a) a linear model; (b) a linear beam model with second order terms; (c) a fully non-linear model. Tip masses at the end of the blades emphasize the coupling between the dynamic and elastic terms. The shape functions are chosen in order to reduce the mathematical model, so that only the first bending mode of the beam is taken into account. The resulting equations of motion have five degrees of freedom and illustrate linear, non-linear and time-dependent terms in a very transparent way. Although neither gyroscopic effect due to rotor angular vibrations nor higher blade mode shapes are considered in the analysis, the equations of motion of the rotor–blades system are still general enough for the purpose of the work: validation of different linear and non-linear models with time dependent (periodic) coefficients. Experiments are carried out in the time and frequency domains while the rotor operates with different constant angular velocities.

In this paper the feasibility of actively suppressing rotor and blade vibration via shaft-based actuation is studied. A mathematical model is derived, taking into account the special dynamical characteristics of coupled rotor-blade systems, such as centrifugal stiffened blades and parametric vibration modes. An investigation of controllability and observability shows that if the blades are properly mistuned, it is possible to suppress shaft as well as blade vibration levels by using only shaft-based actuation and sensing; though, in tuned bladed systems, shaft as well as blade actuation and sensing are required. In order to cope with the time-variant dynamics of the coupled rotor-blade system, a periodic time-variant modal controller is designed, implemented, and experimentally tested. A test rig built by four flexible blades is specially designed for this purpose. The rig is equipped with six electromagnetic actuators and different types of sensors (eddy-current displacement transducers, acceleration transducers, and strain gages) with the aim of monitoring and controlling shaft and blade vibration levels. Two different actively controlled rotor-blade system configurations are considered in the present study: (i) a tuned bladed rotor, controlled with help of actuators attached to the rotating blades and shaft-based actuators; (ii) a deliberately mistuned bladed rotor controlled only via shaft-based actuation. Experimental tests are carried out for both configurations. Some experimental problems regarding control implementation are identified and discussed, especially when the controller order and the number of actuators in the centralized control scheme become too high; though, for the mistuned bladed rotor controlled by using only shaft-based actuation, the controller works well.

Fluid film forces are generated in hydrostatic journal bearings by two types of lubrication mechanisms: the hydrostatic lubrication in the bearing recesses and hydrodynamic lubrication in the bearing lands, when operating in rotation. The combination of both lubrication mechanisms leads to hybrid journal bearings (HJB). When part of hydrostatic pressure is also dynamically modified by means of hydraulic control systems, one refers to the active lubrication. The main contribution of the present theoretical work is to show that it is possible to reduce cross-coupling stiffness and increase the direct damping coefficients by means of the active lubrication, what leads to rotor-bearing systems with larger threshold of stability.

The main objective of this research project is the investigation of multirecess hydrostatic journal bearings with active hybrid (hydrostatic and hydrodynamic) lubrication. This paper gives a theoretical contribution to the modeling of this kind of bearing, combining computational fluid dynamics and control techniques. The feasibility of influencing the dynamic fluid film coefficients (stiffness and damping) by means of a controllable fluid injection into opposed bearing recesses is investigated. By controlling the pressure and flow injection using servo control systems, it is possible to obtain significant modifications of active hybrid forces, which can be useful while reducing vibration and stabilizing rotating machines.

Passive magnetic bearings are known due to the excellent characteristics in terms of friction and no requirement of additional energy sources to work. However, passive magnetic bearings do not provide damping, are not stable and, depending on their design, may also introduce magnetic eccentricity. Such magnetic eccentricities are generated by discrepancies in magnet fabrication. In this framework the main focus of the work is the theoretical as well as experimental investigation of the non-linear dynamics of a rotor-bearing system with strong emphasis on the magnetic eccentricities and non-linear stiffness. In this investigation passive magnetic bearings using axially-aligned neodymium cylinder magnets are investigated. The cylinder magnets are axially magnetised for rotor as well as bearings. Compared to bearings with radial magnetisation, the magnetic stiffness of axially-aligned bearings is considerably lower, nevertheless they allow for asymmetric stiffness mounting, and it could be beneficial for rotor stabilization. A theoretical model is proposed to describe the non-linear rotor-bearing dynamics. It takes into account non-linear behaviour of the magnetic forces and their interaction with a multi-body system composed of rigid rotor and flexible foundation. The magnetic eccentricities of the shaft magnets are modelled using the distances (amplitudes) and directions (phase angles) between the shaft axis and the centre of the magnetic fields generated. A perturbation method, i.e. harmonic balancing, is used in order to evaluate the frequency response of the non-linear system. The experimental validation of the model is carried out using a dedicated rotor-bearing system set-up. The test set-up consists of a vertical rigid shaft and disc supported by two passive magnetic bearings using axially-aligned neodymium cylinder magnets. The magnetic bearing housings are flexibly supported, allowing horizontal motions. The housings are connected to each other by means of elastic beams. The shaft is free in one end and coupled to a DC motor on the other by means of a flexible coupling. On the free end a disc is attached where imbalances and gyroscopic effect can be generated. Comparison between theory and experiment shows high level of resemblance, which validates the theoretical model and the explanations for the quasi-static and dynamic responses. The magnetic eccentricities and mass imbalance effects are clearly detected and distinguished.

Gas bearings have inherent dynamics that gives rise to low damping and potential instability at certain rotational speeds. Required damping and stabilization properties can be achieved by active flow control if bearing parameters are known. This paper deals with identification of parameters in a dynamic model of an active gas bearing and subsequent control loop design. A grey box model is determined based on experiments where piezo actuated valves are used to perturb the journal and hence excite the rotor-bearing system. Such modelling from actuator to output is shown to efficiently support controller design, in contrast to impact models that focus on resonance dynamics. The identified model is able to accurately reproduce the lateral dynamics of the rotor-bearing system in a desired operating range, in this case around the first two natural frequencies. The identified models are validated and used to design a model-based controller capable of improving the damping of the gas bearing. Experimental impact responses show an increase in damping by a factor nine for the investigated conditions.

This paper presents the results obtained from a research work investigating the performance of different Adaptive models developed to predict excitation forces on a dynamically loaded flexible structure. For this purpose, a flexible structure is equipped with acceleration transducers at each degree of freedom and a force transducer for validation and training. The models are trained using data obtained from applying a random excitation force on the flexible structure. The performance of the developed models is evaluated by analyzing the prediction capabilities based on a normalized prediction error. The frequency domain is considered to analyze the similarity of the frequencies in the predicted and the original force signal. For a selection of the best models, a more advanced performance analysis is carried out. This includes application of the trained models to deterministic and non-deterministic excitation forces with different excitation frequencies and amplitudes. Additionally, the influence of the sampling frequency and sensor location on the model performance is investigated. The results obtained in this paper show that most data mining approaches can be used, when a certain degree of inaccuracy is accepted. Furthermore, the comparison study points out that the transducer location is crucial for the model performance. However, there exists no general solution for the final selection of models.Copyright © 2014 by ASME

Corrugated foils are utilized in air foil bearings to introduce compliance and damping thus accurate mathematical predictions are important. A corrugated foil behaviour is investigated experimentally as well as theoretically. Theexperimental investigation is performed by compressing the foil, between two parallel surfaces, both statically and dynamically to obtain hysteresis curves. The theoretical analysis is based on a two dimensional quasi static FEmodel, including geometrical non-linearities and Coulomb friction in the contact points and neglects the foil mass. A method for implementing the friction is suggested. Hysteresis curves obtained via the FE model are compared to the experimental results obtained. Good agreement is observed in the low frequency range and discrepancies for higher frequencies are thoroughly discussed.

In recent years, a continuous research effort has transformed the conventional tilting-pad journal bearing (TPJB) into a mechatronic machine element. The addition of electromechanical elements provides the possibility of generating controllable forces over the rotor as a function of a suitable control signal. Such forces can be applied in order to perform parameter identification procedures “in situ,” which enables evaluation of the mechanical condition of the machine in a noninvasive way. The usage of a controllable bearing as a calibrated shaker requires obtaining the bearing specific frequency dependent calibration function, i.e., the transfer function between control signal and force over the rotor. This work presents a theoretical model of the calibration function for a TPJB with active lubrication. The bearing generates controllable forces by injecting pressurized oil directly into the bearing clearance. The injected flow is controlled by means of a servovalve. The theoretical model includes the dynamics of the hydraulic system using a lumped parameter approach, which is coupled with the bearing oil film using a modified form of the Reynolds equation. The oil film model is formulated considering an elastothermohydrodynamic lubrication regime. New contributions to the mathematical modeling are presented, such as the inclusion of the dynamics of the hydraulic pipelines and the obtention of the bearing calibration function by means of harmonic analysis of a linearized form of the controllable bearing constitutive equations. The mathematical model is used to study the relevance and effects of different parameters on the calibration function, aiming at providing general guidelines for the active bearing design. Finally, experimental results regarding the calibration function and the usage of the studied bearing as a calibrated shaker provide insight into the possibilities of application of this technology.

Tilting-pad journal bearings with multiple orifices hybrid lubrication are analysed by means of a thermohydrodynamic theory. Adiabatic boundary conditions are adopted, and a bidimensional model is used to represent the fluid flow behavior in the bearing gap. The influence of operational conditions on the temperature distribution, when different bore arrangements in the pads are used, is discussed and compared to the theoretical values coming from the conventional hydrodynamic case (without radial oil injection).

Tilting pad journal bearings (TPJB) using multiple orifice hybrid lubrication are analyzed applying a thermohydrodynamic (THD) theory. Adiabatic boundary conditions are adopted, and a two-dimensional model is used to represent the fluid flow behavior in the bearing gap. The influence of operational conditions on the temperature distribution and on the bearing load capacity is discussed and compared to theoretical and experimental values for a conventional hydrodynamic case (without radial oil injection). To improve the cooling effects, as well as rotor attitudes, the best location for orifices is the area near the pad edges.

This paper gives a theoretical treatment of the problem of journal bearings modeling connected to electronic oil injection into the bearing gap. The feasibility of influencing the static behavior of hydrodynamic forces by means of such oil injection is investigated. The lubricant is injected into the bearing gap by two mechanisms of lubrication: the conventional hydrodynamic lubrication and through orifices distributed along the bearing surface (active lubrication in the radial direction). By controlling the pressure of the oil injection, it is possible to get large variations in the active hydrodynamic forces; such effects could be useful for reducing vibrations in rotating machines.

This paper presents an active control strategy based on the periodic time varying linear quadratic regulator problem to attenuate the vibration of a coupled rotor-blade system via shaft actuation. For this system, a periodic modal decomposition method is applied to decouple part of the system dynamics, which ultimately provides a reduced order model. Then, a periodic LQR controller obtained as the solution of a periodic Riccati differential equation is designed. For comparison purpose, three others controllers are also designed. A proportional derivative controller, an LQR controller based on a single LTI model, and an LQR controller based on multiple LTI models. Numerical simulations show the superior performance of the proposed technique.

The servo valve input signal and the radial injection pressure are the two main parameters responsible for dynamically modifying the journal oil film pressure and generating active fluid film forces in controllable fluid film bearings. Such fluid film forces, resulting from a strong coupling between hydrodynamic, hydrostatic and controllable lubrication regimes, can be used either to control or to excite rotor lateral vibrations. An accurate characterization of the active oil film forces is of fundamental importance to elucidate the feasibility of applying the active lubrication as non-invasive perturbation forces, or in other words, as a “calibrated shaker”, to perform in-situ rotordynamic tests. The main original contributions of this paper are three: a) the experimental characterization of the active fluid film forces generated in an actively-lubricated tilting-pad journal bearing in the frequency domain and the application of such a controllable bearing as a calibrated shaker aiming at determining the frequency response function (FRF) of rotordynamic systems; b) experimental quantification of the influence of the supply pressure and servo valve input signal on the FRF of rotor-journal bearing systems; c) experimental indication of how small such active fluid film forces (perturbation forces) should be, in order to perturb the rotor-journal bearing system without significantly changing its dynamic characteristics. To validate the experimental procedure and results obtained via actively-lubricated bearing, similar experimental tests are carried out using an electro-magnetic shaker. Very good agreements between the two experimental approaches are found. Maximum values of the main input parameters, namely servo valve voltage and radial injection pressure, are experimentally identified/suggested with the objective of obtaining non-invasive perturbation forces.Copyright © 2013 by ASME

This paper gives a theoretical and experimental contribution to the problem of active modification of the dynamic coefficients of tilting-pad journal bearings, aiming to increase the damping and stability of rotating systems. The theoretical studies for the calculation of the bearing coefficients are based on the fluid dynamics, specifically on the Reynolds equation, on the dynamics of multibody systems and on some concepts of the hydraulics. The experiments are carried out by means of a test rig specially designed for this investigation. The four pads of such a bearing are mounted on four flexible hydraulic chambers which are connected to a proportional valve. The chamber pressures are changed by means of the proportional valve, resulting in a displacement of the pads and a modification of the bearing gap. By changing the gap, one can adjust the dynamic coefficients of the bearing. With help of an experimental procedure for identifying the bearing coefficients, theoretical and experimental results are co...

The possible contact between rotor and stator can for some cases be considered a serious malfunction that may lead to catastrophic failure. Rotor rub is considered a secondary phenomenon caused by a primary source that leads to a disruption of the normal operational condition. It arises from sudden mass unbalance, instabilities generated by aerodynamic and hydrodynamic forces in seals and bearings among others. The contact event gives rise to normal and friction forces exerted on the rotor at impact events. The friction force plays a significant role by transferring some rotational energy of the rotor to lateral motion, impacting the stator. This event results in persistent coupled lateral vibration of the rotor and stator. This paper proposes a new unconventional backup bearing design in order to reduce the rub related severity in friction. The idea is to utilize pin connections that center the rotor during impacts. In this way, the rotor is forced to the center and the lateral motion is mitigated. The four pins are passively adjustable, which allows the clearance to be customized. A mathematical model has been developed to capture phenomena arising from impact for the conventional backup bearing (annular guide) and for the new disk-pin backup bearing. For the conventional annular guide setup, it is reasonable to superpose an impact condition to the rub, where the rotor spin energy can be fully transformed into rotor lateral movements. Using a nonideal drive, i.e., an electric motor without any kind of velocity feedback control, it is even possible to almost stop the rotor spin under rubbing conditions. All the rotational energy will be transformed in a kind of “self-excited” rotor lateral vibration with repeated impacts against the housing. The vibration of the housing is coupled through the interaction force. The experimental and numerical analysis shows that for the conventional annular guide setup, the rotational energy is fully transformed into lateral motion and the rotor spin is stopped. However, by employing the new disk-pin design the analysis shows that the rotor at impact is forced to the center of the backup bearing and the lateral motion is mitigated. As a result of this, the rotor spin is kept constant.

Along with traditional oil lubrication, increasing demand for high-speed applications has renewed attention to gas bearings technology. Traditional aerostatic and aerodynamic gas lubrication has been widely used in a variety of applications, ranging from high-speed spindles to micro and meso-scale turbo-machinery. The present paper deals with experimental rotordynamic testing of a flexible rotor supported by hybrid aerostatic-aerodynamic gas journal bearing equipped with an electronic radial air injection system. From a rotordynamic point of view there are two phenomena that limit the widespread of traditional gas lubrication: 1) Low damping makes operation across critical speed dangerous, as even low level of unbalance can generate large vibration responses. This is especially problematic for gas bearing applications, which often operate in the supercritical region. Moreover, 2) An upper bound to supercritical operation is determined by the appearance of subsynchronous whirl instability. Due to the sudden increase in amplitude with respect to speed, this most often corresponds to the maximal attainable rotational speed of the system. Postponing the onset speed of instability poses therefore one of the greatest challenges in a high-speed gas bearing design. A great deal of research is devoted to attack such issues, where most propose passive designs such as compliant foil bearings, tilting pad and flexure pivot gas bearings. These solutions proved to be effective in improving static and dynamic properties of the bearings, however issues related to the manufacturing and accuracy of predictions has so far limited their applications. Another drawback is that passive bearings offer a low degree of flexibility, meaning that an accurate optimization is necessary for each application. The developed prototype active bearing offers several promising performance enhancements. Synchronous vibrations can be effectively addressed ensuring safe operation across the critical speeds; whirling instability is suppressed; intervening on the software, rather than the hardware can modify the response of the system. Implementing active lubrication adds however a considerable number of parameters and variables. The performance of a good control system lays most importantly on a good choice of control gains, which in general are different depending on the goal of the controller. Optimum tuning of the control loop is addressed experimentally, showing dependency on the supply pressure and, less prominently, the rotational velocity.Copyright © 2012 by ASME

This paper gives an original theoretical and experimental contribution to the issue of reducing force estimation errors, which arise when applying Active Magnetic Bearings (AMBs) with pole embedded Hall sensors for force quantification purposes. Motivated by the prospect of increasing the usability of AMBs by embedding Hall sensors instead of mounting these directly on the pole surfaces, force estimation errors are investigated both numerically and experimentally. A linearized version of the conventionally applied quadratic correspondence between measured Hall voltage and applied AMB force is suggested and investigated. A finite element (FE) model is constructed to study force error behavior as a function of rotor offset. The investigation confirms that the magnitude of the force error is dependent on how well the rotor is centered in the AMB. Furthermore, below a rotor offset corresponding to ∼20% of the nominal air gap the force estimation error is found to be reduced by the linearized force equation as compared to the quadratic force equation, which is supported by experimental results. Additionally the FE model is employed in a comparative study of the force estimation error behavior for pole embedded and pole surface mounted Hall sensors. It is shown that in a given range of bias currents and rotor offsets, pole embedded and surface mounted Hall sensors perform equally well for the four pole heteropolar flux-split radial AMB under investigation. Furthermore, frequency dependence of the Hall sensor sensitivity factors is investigated, and found to be non-existing, hence static calibration of Hall sensors is sufficient, even for dynamic testing purposes.Copyright © 2012 by ASME

The relevance of calculating accurately the oil film temperature build up when modeling tilting-pad journal bearings is well established within the literature on the subject. This work studies the feasibility of using a thermal model for the tilting-pad journal bearing which includes a simplified formulation for inclusion of the heat transfer effects between oil film and pad surface. Such simplified approach becomes necessary when modeling the behavior of tilting-pad journal bearings operating on controllable lubrication regime. Three different simplified heat transfer models are tested, by comparing their results against the ones obtained from an state of the art tilting-pad journal bearing model, where the heat transfer effects are throughly implemented, as well as against some experimental results from the literature. The results obtained show that the validity of the simplified heat transfer models are strongly dependent on the Reynolds number for the oil flow in the bearing. For bearings operating in laminar regime, the decoupling of the oil film energy equation solving procedure, with no heat transfer terms included, with the pad heat conduction problem, where the oil film temperature is applied at the boundary as a Dirichlet condition, showed a good balance between quality of the results, implementation easiness and reduction in calculation time. For bearings on the upper limit of the laminar regime, the calculation of an approximated oil film temperature gradient in the radial direction, as proposed by Knight and Barrett, delivered the best results.Copyright © 2012 by ASME

This paper presents the latest research results of a project that focuses on normal behavior models for condition monitoring of wind turbines and their components, via ordinary Supervisory Control And Data Acquisition (SCADA) data. In this machine learning approach Adaptive Neuro-Fuzzy Interference System (ANFIS) models are employed to learn the normal behavior in a training phase, where the component condition can be considered healthy. In the application phase the trained models are applied to predict the target signals, e.g. temperatures, pressures, currents, power output, etc. The behavior of the prediction error is used as an indicator for normal and abnormal behavior, with respect to the learned behavior. The advantage of this approach is that the prediction error is widely decoupled from the typical fluctuations of the SCADA data caused by the different turbine operational modes. To classify the component condition Fuzzy Interference System (FIS) structures are used. Based on rules that are established with the prediction error behavior during faults previously experienced and generic rules, the FIS outputs the component condition (green, yellow and red). Furthermore a first diagnosis of the root cause is given. In case of fault patterns earlier unseen the generic rules allow general statements about the signal behavior which highlight the anomaly. Within the current research project this method is applied to 18 onshore turbines of the 2 MW class operating since April 2009. First results show that the proposed method is well suited to closely monitor a large variety of signals, identify anomalies and correctly classify the component condition. The accuracy of the normal behavior models developed is high and small signal behavior changes become recognizable. The result of the automatic analysis is given in graphical and text format. Within the paper examples of real faults are provided, showing the capabilities of the method proposed. The method can be applied both to existing and new built turbines without the need of any additional hardware installation or manufacturers input.

We present a software toolbox that allows to apply continuation methods directly to a controlled lab experiment. This toolbox enables us to systematically explore how stable and unstable steady state periodic vibrations depend on parameters. The toolbox is implemented partly in MATLAB and partly on a dSPACE realtime controller board. Its functionality is tested on a driven mechanical oscillator with a strong impact nonlinearity, controlled with electromagnetic actuators. We show how to tune a controller so that the steady state dynamics of the controlled experiment matches that of the corresponding un-controlled experiment.

This paper presents the research results of a comparison of three different model based approaches for wind turbine fault detection in online SCADA data, by applying developed models to five real measured faults and anomalies. The regression based model as the simplest approach to build a normal behavior model is compared to two artificial neural network based approaches, which are a full signal reconstruction and an autoregressive normal behavior model. Based on a real time series containing two generator bearing damages the capabilities of identifying the incipient fault prior to the actual failure are investigated. The period after the first bearing damage is used to develop the three normal behavior models. The developed or trained models are used to investigate how the second damage manifests in the prediction error. Furthermore the full signal reconstruction and the autoregressive approach are applied to further real time series containing gearbox bearing damages and stator temperature anomalies. The comparison revealed all three models being capable of detecting incipient faults. However, they differ in the effort required for model development and the remaining operational time after first indication of damage. The general nonlinear neural network approaches outperform the regression model. The remaining seasonality in the regression model prediction error makes it difficult to detect abnormality and leads to increased alarm levels and thus a shorter remaining operational period. For the bearing damages and the stator anomalies under investigation the full signal reconstruction neural network gave the best fault visibility and thus led to the highest confidence level.

This work gives an overview of the theoretical and experimental achievements of mechatronics applied to fluid film bearings. Compressible and uncompressible fluids are addressed. Rigid and elastic (deformable) bearing profiles are investigated. Hydraulic, pneumatic, magnetic and piezoelectric actuators are used. The ideas of combining control techniques, informatics with hydrodynamic, thermo-hydrodynamic, elasto-hydrodynamic and thermo-elasto-hydrodynamic lubrication techniques are carefully explored in this paper, considering theoretical as well as experimental aspects. The main goal of using controllable fluid film bearings is to improve the overall machine performance by: controlling the lateral vibration of rigid and flexible rotating shafts; modifying bearing dynamic characteristics, such as stiffness and damping properties; increasing the rotational speed ranges by enhancing damping and eliminating instability problems, for example, by compensating cross-coupling destabilizing effects; reducing start-up torque and energy dissipation in bearings; compensating thermal effects. It is shown that such controllable fluid film bearings can act as “smart” machine components and be applied to rotating and reciprocating machines with the goal of avoiding unexpected stops of plants, performing rotor dynamic tests and identifying model parameters “on site”. Emphasis is given to the controllable lubrication (hybrid and active) applied to different types of oil film bearings under different lubrication regimes, i.e., as tilting-pad journal bearings, multi-recess journal bearings and plain journal bearings. After a comprehensive overview of the theoretical and experimental technological advancements achieved in university laboratories, the feasibility of industrial applications is highlighted, trying to foresee the future trends of such mechatronic devices.

This work is aimed at a theoretical study of the dynamic behavior of a rotor-tilting pad journal bearing (TPJB) system under different lubrication regimes, namely, thermohydrodynamic (THD), elastohydrodynamic (EHD), and hybrid lubrication regime. The rotor modeled corresponds to an industrial compressor. Special emphasis is put on analyzing the stability map of the rotor when the different lubrication regimes are included into the TPJB modeling. Results show that, for the studied rotor, the inclusion of a THD model is more relevant when compared to an EHD model, as it implies a reduction on the instability onset speed for the rotor. Also, results show the feasibility of extending the stable operating range of the rotor by implementing a hybrid lubrication regime.

This work gives an overview about the theoretical and experimental achievements of mechatronics applied to oil film bearings, with the aim of: controlling the lateral vibration of flexible rotating shafts; modifying bearing dynamic characteristics, as stiffness and damping properties; increasing the rotational speed ranges by improving damping and eliminating instability problems, for example, by compensating cross-coupling destabilizing effects; reducing startup torque and energy dissipation in bearings; compensating thermal effects. It is shown that such controllable bearings can act as “smart” components and be applied to rotating machines with the goal of avoiding unexpected stops of plants, performing rotordynamic tests and identifying model parameters “on site”. Emphasis is given to the controllable lubrication (active lubrication) applied to different types of oil film bearings, i.e., as tilting-pad bearings, multirecess journal bearings and plain bearings.

The static and dynamic properties of tilting-pad journal bearings with controllable radial oil injection are investigated theoretically. The tilting pads are modeled as flexible structures and their behavior is described using a three-dimensional finite element framework and linear elasticity. The oil film pressure and flow are considered to follow the modified Reynolds equation, which includes the contribution from controllable radial oil injection. The Reynolds equation is solved using a two-dimensional finite element mesh. The rotor is considered to be rigid in terms of shape and size, but lateral movement is permitted. The servovalve flow is governed by a second order ordinary differential equation, where the right hand side is controlled by an electronic input signal. The constitutive flow-pressure relationship of the injection orifices is that of a fully developed laminar velocity profile and the servovalve is introduced into the system of equations by a mass conservation consideration. The Reynolds equation is linearized with respect to displacements and velocities of the nodal degrees of freedom. When all nodal points satisfy static equilibrium, the system of equations is dynamically perturbed and subsequently condensed to a 2×2 system, keeping only the lateral motion of the rotor. As expected, bearing dynamic coefficients are heavily influenced by the control parameters and pad compliance.

In order to reduce friction losses and vibrations in main engine bearings, fluid film lubrication is combined to controllable radial oil injection. This work evaluates different geometric hybrid bearing configurations and control pressure rules for applying radial oil injection in main engine bearings of internal combustion engines. The conventional hydrodynamic lubrication (CHL) is combined with hydrostatic lubrication which is actively modified by radially injecting oil at controllable pressures, through orifices circumferentially located around the bearing surface. The behaviour of a main bearing of a medium size combustion engine, operating with radial oil injection and with four different control strategies is analyzed, giving some insights into the minimum fluid film thickness (OFT) and reduction of viscous friction losses. The behaviour of such parameters is compared to the case when the bearing operates with conventional hydrodynamic lubrication.Copyright © 2010 by ASME

In this work, the steps involved for the modelling of a reciprocating linear compressor are described in detail. The dynamics of the mechanical components are described with the help of multibody dynamics (rigid components) and finite elements method (flexible components). Some of the mechanical elements are supported by fluid film bearings, where the hydrodynamic interaction forces are described by the Reynolds equation. The system of nonlinear equations is numerically solved for three different restrictive conditions of the motion of the crank, where the third case takes into account lateral and tilting oscillations of the extremity of the crankshaft. The numerical results of the behaviour of the journal bearings for each case are presented giving some insights into design parameters such as, maximum oil film pressure, minimum oil film thickness, maximum vibration levels and dynamic reaction forces among machine components, looking for the optimization and application of active lubrication towards vibration reduction.

The static and dynamic properties of tilting-pad journal bearings with controllable radial oil injection are investigated. The tilting pads are modelled as flexible structures and their dynamics are described using a three dimensional finite element framework and linear elasticity. The oil film pressure and flow are considered to follow the modified Reynolds equation, which includes the contribution from controllable radial oil injection. The Reynolds equation is solved using a two dimensional finite element mesh. The rotor is considered to be rigid. The servo-valve flow is governed by a second order ordinary differential equation, where the right hand side is controlled by an electronic input signal. The constitutive flow pressure relationship of the injection nozzles is that of a fully developed laminar velocity profile and the servo-valve is introduced into the system of equations by a volume conservation consideration. The Reynolds equation is linearized with respect to displacements and velocities of the nodal degrees of freedom. When all nodal points satisfy the static equilibrium condition, the system of equations is dynamically perturbed and subsequently condensed to a 2 by 2 system, keeping only the lateral motion of the rotor. As expected, rotor stability is heavily influenced by the control parameters.Copyright © 2008 by ASME

This work describes in details the steps involved within the mathematical modelling of multibody systems (rigid and flexible) interconnected via controllable thin fluid films. The dynamics of the mechanical components are described with help of multibody dynamics and finite element method. In this paper, the methodology is applied to reciprocating machines such as hermetic reciprocating compressors and internal combustion engines. In previous studies [1], it has been shown that for a light duty vehicle, the friction losses may reach until 48% of the total energy consumption of an engine and from that, almost 30% are coming from bearings and crankshaft. Therefore, considering that the dynamics of the fluid films in the journal bearings can be actively controlled by means of different types of actuators, allowing significant reduction of wear and vibrations, one of the aims of this paper is to study the feasibility of applying active lubrication to the main journal bearings of reciprocating machines. In this framework the paper gives a theoretical contribution to the combined fields of fluid-structure interaction and active vibration control. The hydrodynamic pressure distribution for an active lubricated finite journal bearing dynamically loaded can be calculated by numerically solving the modified Reynold’s equation [2], by means of finite-difference method and integrated over the pressure area in order to obtain the dynamic reaction forces among components. These forces are strongly nonlinear and dependent on the relative kinematics of the system. From the point of view of active lubrication and specifically considered the case of a dynamically loaded journal bearing, the injection pressure should be controlled in the time domain. However, taking into account that the pressures and reaction forces in a reciprocating machine have a cyclic behaviour, the fluid film thickness of the main bearings may be modified by controlling the oil pressure injection, depending on the crank angle and the load bearing condition. It can be mentioned that the pressure and flow may be controlled by mechanical cam systems, piezoelectric nozzles [3] [4] or servovalves [5] [6], therefore, an adequate control strategy has to be defined. The fluid film forces are coupled to the set of nonlinear equations that describes the dynamics of the mechanical system. Such a set of equations is numerically solved giving some insights into the following parameters: a) maximum fluid film pressure, b) minimum fluid film thickness, c) maximum vibration levels and d) viscous frictional forces. The behaviour of such parameters is investigated when the system operate with conventional hydro-dynamic lubrication, passive hybrid lubrication and controlled hybrid lubrication.Copyright © 2008 by ASME

A three-dimensional thermoelastohydrodynamic model is applied to the analysis of tilting-pad bearings with spherical pivots and equipped with deep recesses in the high-pressure regions. A potential for a 10–20% reduction in the friction loss compared to conventional plain bearing pads is documented. Design suggestions minimizing the power loss are given for various length-to-width ratios. The tilting angle in the sliding direction is more sensitive to correct positioning of the pivot point than conventional bearing pads. Improving the performance by equipping a tilting-pad bearing with a deep recess therefore requires accurate analysis and design of the bearing. Similarly, a high sensitivity perpendicular to the sliding direction suggests that this method of reducing friction is more feasible when using line pivots or spring beds than when using spherical pivots for controlling the tilting angle.

Gas bearings are popular for their high speed capabilities, low friction and clean operation, but require low clearances and suffer from poor damping properties. The poor damping properties cause high disturbance amplification near the natural frequencies. These become critical when the rotation speed coincides with a natural frequency. In these regions, even low mass unbalances can cause rub and damage the machine. To prevent rubbing, the variation of the rotation speed of machines supported by gas bearings has to be carefully conducted during run-ups and run-downs, by acceleration and deceleration patterns and avoidance of operation near the critical speeds, which is a limiting factor during operation, specially during run-downs. An approach for reducing the vibrations is by feedback controlled lubrication. This paper addresses the challenge of reducing vibrations in rotating machines supported by gas bearings to extend their operating range. Using H∞-design methods, active lubrication techniques are proposed to enhance the damping, which in turn reduces the vibrations to a desired safe level. The control design is validated experimentally on a laboratory test rig, and shown to allow safe shaft rotation speeds up to, in and above the two first critical speeds, which significantly extends the operating range.

It is often not feasible or even impossible to identify a plant in open loop. This might be because the plant contains unstable poles, or it is simply too expensive to remove the plant from its intended operation, among other possibilities. There are several methods for identifying a plant in closed loop [4], and one such method is the Hansen scheme [1]. Standard identification using Hansen scheme demands generating the identification signals indirectly. In this paper it is instead proposed to use the relationship between the Youla factorization of a plant and its stabilizing controller to directly measure the signals used for identification. A simulation example and identification of a gas bearing is given to show the method in action. Rotors supported by controllable gas bearings are open loop stable systems. However as the rotational speed is increased feedback control is necessary in order to keep the system stable. Furthermore because the dynamics of such a system depends on the rotational speed it is needed to conduct an identification while the system is part of a closed loop scheme. The authors believe the paper able to contribute towards a simpler and more direct way of identifying closed loop plants using Hansen scheme.

This is Part I of a two-part series of papers describing the effects of high pressure injection pockets on the operating conditions of tilting-pad thrust bearings. A numerical model based on the Reynolds equation is developed extending the three dimensional thermo-elasto-hydrodynamic (TEHD) analysis of tilting-pad thrust bearings to include the effects of high pressure injection and recesses in the bearing pad. The model is applied to the analysis of an existing bearing of large dimensions and the influence of the pocket is analyzed. It is shown that a shallow pocket positively influences the performance of the bearing as it has characteristics similar to those of a parallel step bearing.Copyright © 2006 by ASME

Journal bearings under active lubrication regime are controlled by servo valves and well-tuned feedback control laws. The servo valves dynamically modify the journal pressure distribution generating active oil film forces. Such forces are dependent on the following parameters: Sommerfeld number, bearing pre-load factor, orifice diameter, excitation frequency, feedback control gain and dynamic parameters of the servo valves, i.e. their natural frequencies, damping factors and pressure-flow coefficients. The theoretical and experimental characterization of such active forces is the main focus and the main contribution of this work. The theoretical analyzes are based on the Modified Reynolds’ equation for active lubrication. The experimental analyzes are carried out by using a special test rig, designed to investigate the behavior of tilting-pad bearings (TPJB) under active lubrication. Relative good agreement between theoretical and experimental behavior of the active oil film forces as a function of the excitation frequency and control gain is achieved. Limitations of the characterization procedure is critically discussed in order to explain some discrepancies between theoretical and experimental results.Copyright © 2005 by ASME