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Four different magnet bearing configurations, with varying number of combined radially and axially magnetized rings, are studied both experimentally and using the modeling framework MagTense. First, the optimal vertical position of the radially magnetized ring is determined using the numerical model. Then, the model is validated using experimental data. Finally, we show that a bearing where the rotor and the stator each consist of a single ring of axially magnetized magnets and a single ring of radially magnetized magnets has the smallest radial force per axial lift force at an axial air gap of 1 mm. For this bearing, the ratio of the radial force per axial lift here is 0.383 times that of the same bearing without the radially magnetized ring.

Rotor dynamic force coefficients of gas seals strongly depend on the machine operational conditions. These force coefficients influence the overall dynamical response and modal properties of machines, consequently defining the machine vibration levels. Accurate estimations of the rotor dynamic coefficients are required for designing machines with low vibration amplifications and well-defined stability margins throughout the operational range. Experimental methods applied to test benches are used to validate such force coefficients and they normally rely on (i) the quality of the measurements and (ii) the assumption that the mathematical model is able to capture the whole system dynamics. If relevant dynamical contributions in a system are neglected by the mathematical model, the contribution will erroneously be concluded to originate from the seal being tested. The theoretical and experimental investigation in this paper focuses on quantifying and qualifying the effect of neglected system dynamics modelling on the estimation of seals force coefficients and stability margins. The in-situ identification of seal forces shows that the direct stiffness, cross-coupling stiffness, and direct damping coefficient estimations for a gas seal with high preswirl are statistically significantly affected by the baseline model. Nevertheless, the baseline model leads to small deviations of the seal force coefficient estimations. The prediction accuracy of stability margins is found to be more influenced by the baseline model describing the system dynamics than by the deviations between the seal force coefficient estimations.

This paper gives a theoretical and experimental contribution to the problem of operational modal analysis (OMA) applied to linear time-periodic (LTP) systems, with focus on rotor-blade coupled dynamics. To evaluate advantages, disadvantages, and challenges a dedicated test rig is designed and fully instrumented allowing for traditional experimental modal analysis (EMA) as well as OMA. The test rig is composed of four flexible blades attached to a rigid rotating disc. To vary the strength of the dynamic coupling between rotor and blades, tip masses are mounted at the free ends of the blades. Multibody dynamics techniques are used to build a mathematical model for describing the interaction between rotor and blades. A total of six degrees of freedom — two linear displacements of the rotating disc and four linear translations of the tip masses at the end of the flexible blades – are measured using contactless displacement sensors. All six degrees of freedom are also electromagnetically excited avoiding mechanical contact. The Ibrahim time domain (ITD) identification technique is applied to the six measured signals in order to extract the fundamental and parametric natural frequencies and damping ratios of the rotor-blade system. The experimental results from the ITD identification are validated against the mathematical model and traditional EMA. The influence of angular velocity on the natural frequencies is investigated. It can be concluded, that three types of vibrational modes can be detected depending on sensor location: i) coupled modes dominated by the disc movement, ii) coupled modes dominated by the blade tip masses and iii) pure blade tip mass modes. Measuring the rotor displacements allows for detecting the parametric vibrations of the blade tip masses as well as fundamental vibrations of the rotor disc, while measuring blade displacements allows for detecting the parametric vibrations of the rotor disc as well as fundamental vibrations of the blade tip masses, influenced by the rotational stiffening. The advantages of OMA are clear since only output signals are needed for the analysis. Challenging is though the identification and classification of measured parametric modes, especially those related to the blade tip masses where a high modal density is always expected and those in frequency ranges where veering effect occurs. To experimentally obtain accurate dynamic responses for the blades via EMA when the rotor-blade system is excited by electromagnetic actuators acting on the hub, the electromagnetic forces have to be projected in the direction of the blades’ degrees-of-freedom, using a similar definition of generalised forces. For OMA such a procedure is bypassed due to the assumption of unknown excitation forces.

The study of rotating machines stands out in the context of systems and structures due to the significant number of typical phenomena that may be present during the operation of such equipment. Rotating systems play an important role in industrial environments, with a wide range of application, namely, pumps, turbines, generators and compressors, and turbochargers, among others. Given this context, tilting pad journal bearings (TPJBs) can offer considerable stability to the rotor system due to its damping and stiffness characteristics, especially under high rotation speeds, whereas journal bearings with a fixed geometry may be subjected to fluid-induced instability. Consequently, thermal effects are significant in rotating machines to evaluate the behavior of hydrodynamic bearings due to the increase in shear effects in the lubricant caused by the rotation speed, leading to excessive temperature increase under specific operational conditions and, therefore, significant changes in lubricant viscosity. These effects may influence the dynamic characteristics of bearings, affecting the dynamic response of the entire system. From this perspective, a specific test rig was designed to perform the characteristic uncoupled motion of the shaft when supported by this kind of bearing. Experimental tests were directly compared with numerical models, showing promising results. In addition, the thermohydrodynamic (THD) lubrication was numerically solved by a finite volume method, considering an approach for a 3D THD model and the pivot flexibility.

This paper presents an active control strategy, based on a time-varying linear–quadratic optimal control problem, to attenuate the tip vibration of a two-dimensional coupled rotor-blade system whose dynamics is periodic. First, a periodic full-state feedback controller based on the linear–quadratic regulator (LQR) problem is designed. If all the states are not available for feedback, then an optimal periodic time-varying estimator, using the Kalman–Bucy filter, is computed. Both the Kalman filter gain and the LQR gain are obtained as the solution of a periodic Riccati differential equation (PRDE). Together, these gains provide the observer-based linear–quadratic–Gaussian (LQG) controller. An algorithm to solve the PRDE is also presented. Both controller designs ensure closed-loop stability and performance for the linear time-varying rotor-blade equation of motion. Numerical simulations show that the LQR and the LQG controllers are able to significantly attenuate the rotor-blade tip vibration.

This article deals with the design of a test bench and its dynamic testing. The test bench is composed of a rotor levitated by radial active magnetic bearings and axially by a permanent magnet bearing. The rotor housing is mounted in a passive gimbal and the foundation can translate and rotate to emulate a vehicle accelerating, manoeuvring, and subject to outer perturbations such as waves. The work is motivated by a lack of publically available experimental designs and experimental data for magnetically suspended flywheels on moving foundations. The article will describe the test rig design and subcomponents, present experimental results, and afterwards the test rig will be used to experimentally validate a mathematical model governing the dynamics of the system. It is found that the model successfully captures the dynamics of rotor and housing both in a gimballed and non-gimballed flywheel energy storage system when subject to accelerations of the foundation. The simulated maximum housing accelerations are found to deviate 2% from the experiments while the maximum rotor displacements are found to deviate 37%, but only for rotor displacements larger than 100 μm where the linear approximation to the magnetic forces is no longer accurate.

A new type of flywheel energy storage system uses a magnetic suspension where the axial load is provided solely by permanent magnets, whereas active magnetic bearings are only used for radial stabilization. This means that the permanent magnet bearing must provide all the axial damping. Furthermore, it must have as low a negative radial stiffness as possible to reduce the workload on the radial active magnetic bearings. Many different mathematical models for determining force, stiffness, and damping of permanent magnet bearings are available in the literature. This work will further develop the most applicable analytical and numerical methods in order to make them directly implementable for designing permanent magnet thrust bearings for flywheel energy storage systems. The outcome is a fast and efficient method for determining force, stiffness, and damping when the bearing setup contains magnetic materials with relative permeability higher than one as well as when it does not. The developed method is validated against numerical and experimental results with good agreement.

The vibro-impact response of a single-degree of freedom model with the coefficient of restitution is analyzed using pointwise mapping and a standard averaging combined with non-smooth transformations. Experimental data are taken from a cantilever beam with attached mass and unilateral constraint submitted to different gap configurations and levels of excitation. Numerical simulations are used to reproduce empirical observations to a certain extent and validate theoretical predictions. Investigations on the coefficient of restitution show its dependence on the forcing frequency and pre-contact velocity. The effect of gap variations due to sliding of the constraint during frequency sweep is analyzed experimentally.

Recently research into active gas bearings has had an increase in popularity. There are several factors that can make the use of gas bearings favourable. Firstly gas bearings have extremely low friction due to the usage of gas as the lubricant which reduce the needed maintenance. Secondly gas bearings is a clean technology which makes it possible to use for food processing, air condition and applications with similar requirements. Active gas bearings are therefore useful for applications where downtime is expensive and dirty lubricants such as oil are inapplicable. In order to keep as low downtime as possible it is important to be able to determine when a fault occurs. Fault diagnosis of active gas bearings is able to minimize the necessary downtime by making certain the system is only taken offline when a fault has occurred. Usually industry demands the removal of any sensor redundancy in systems. This makes it impossible to isolate faults using passive fault diagnosis. Active fault diagnosis methods have been shown able to isolate faults when there is no sensor redundancy. This makes active fault diagnosis methods relevant for industrial systems. It is in this paper shown possible to apply active fault diagnosis to diagnose parametric faults on a controllable gas bearing. The fault diagnosis is based on a statistical detector which is able to quantify the quality of the diagnosis scheme.

3D CFD simulations are performed to resolve the leakage rates, axial pressure drops, and dynamic force coefficients of smooth annular seals. Simulations are performed for several, simple annular seal geometries subject to water and gas flows in order to verify and validate the open-source CFD library OpenFOAM for such applications. Rotordynamic force coefficients are determined using the whirling-rotor method. Accuracy of CFD-predicted results is assessed through direct comparison with published experimental data and bulk-flow model predictions. For the three water seals analyzed, the open-source, CFD-predicted dynamic coefficients closely agree with their experimental counterparts. The k- $$\omega $$ family of turbulence models is shown to outperform the k- $$\epsilon $$ models. For the gas seals, the open-source CFD solver is shown to be unsatisfactory in predicting damping coefficients and simulations are alternatively performed with ANSYS Fluent.

In rotating machines, tilting pad bearings (TPJBs) are widely used due to their enhanced design stability in comparison with alternative bearing configurations. In the present research, a thermohydrodynamic (THD) lubrication model for a tilting pad bearing with injection is developed. Within the THD lubrication model, the lubricant is also fed into each pad through a central injection orifice. The effects on the lubricant oil film pressures and temperatures and subsequent influences on the system dynamic behavior were investigated. Another important factor considered within the THD model is associated with the flexibility of the pivot of each pad and its impact on the behavior of the bearing when subjected to direct lubricant injection. The numerical model results are compared with the measurements of the rotor equilibrium position within the bearing and the pad temperatures obtained from the test rig designed specifically for the TPJB geometry with and without central pad injection. Pad temperatures are measured before the injection orifice position and after it. Agreement between experimental and model-predicted results provides confidence in the model’s ability to evaluate the dynamic behavior of the bearing and its effect on the complete system through the stiffness and damping coefficients

A large part of keeping a process running safely is ensuring that all equipment is functioning well. To do so, routine preventive maintenance needs to be conducted. This paper investigates the feasibility of a functional modeling method-Multilevel Flow Modeling (MFM) extension with roles for modeling of mechanical components functionalities. The method is applied on a case study of a test bench. Consequently, it studies the effects of physical structure failures on functions propagated upwards to objectives. This work can benefit in making the maintenance plan with further experimental study for linking condition monitoring data with the extended MFM models.

This paper considers the problem of real-time estimation of ground vehicle mass, based on longitudinal dynamics. Compared to previous contributions, this work relies on basic data from the vehicle controller area network bus. This allows for application to a very broad range of vehicles, by not using vehicle specific parameters such as current gear indicator and engine torque. The current gear ratio is predicted based on the engine and vehicle speeds. The engine torque is modelled by the mass air flow rate of the engine. The road gradient is estimated with an inertial measurement unit, with fusion of gyroscope and accelerometer using a complementary filter. To handle prolonged dynamic conditions affecting the performance of the filter, compensation of external accelerations is made by use of the vehicle speed from the controller area network bus. The mass is estimated with a recursive least squares filter with forgetting factor. The model is validated experimentally with data obtained from test drives with two different petrol powered passenger cars, equipped with a manual transmission and dual clutch automatic transmission respectively. The test drives consist of various driving conditions with different vehicle loads. The resulting mass estimates are generally good for both vehicles with variations within ± 5% of the actual masses.

This work presents a new methodology for the topology optimization of piezoelectric actuators in laminated composite structures with the objective of controlling external perturbation induced by structural vibrations. The linear-quadratic regulator (LQR) optimal control technique is used and the topology optimization is formulated seeking to find the optimum localization of the macro-fiber composite (MFC) active piezoelectric patch by means of the maximization of the controllability index. For the structural model, we propose a simplified MFC/structure interaction model. It is assumed that the MFC is one of the orthotropic material layers with an initial strain arising from the application of an electric potential. This strain acts on the remainder of the structure and its effect is considered analytically. Numerical results show that the proposed MFC structure interaction model presents good agreement with experiments and numerical simulations of models that take into account the electromechanical effect. Results of the actuator location optimization show that the implemented technique improves the structural vibration damping. Results and comparisons are presented for the vibration control strategy using the LQR controller.

Flywheel energy storage systems (FESSs) with active and passive magnetic bearings are generating interest due to their increasing energy-storing potential caused by advances in motor, bearing and fibre-composite technology. Magnetically suspended FESS are used commercially on static foundations while vehicle applications pose challenges due to manoeuvring, impacts, and other outer perturbations and have thus only seen successful experimental application in a few research projects where the FESS has been mounted in a passive gimbal to avoid gyroscopic forces. Although experimentally implemented, a mathematical model is still missing that determines motions and forces when the FESS is suspended in magnetic bearings, gimbal-mounted, and subject to outer perturbations. This work thoroughly describes how to set up a mathematical model that couples the multi-body dynamics of a flywheel rotor, housing, and gimbal-mount with the magnetic forces of the bearings. The model is used to simulate the behaviour of a FESS with and without gimbal and subject to various perturbations. The results demonstrate how the gimbal mount effectively removes gyroscopic forces but introduces other potential challenges such as large rotor and housing displacements due to dynamical interactions between the rotor, active magnetic bearings, housing, and gimbal.

The present contribution is devoted to the study of the influence of shape memory alloys on the dynamic behavior of flexible rotors. In this sense, a suspension composed by pseudoelastic shape memory alloy wires that are connected to a rotor-bearing test rig was designed. To evaluate the performance of the system, both numerical and experimental investigations are carried out. The suspension stiffness can vary, especially in the pseudoelastic region, so that this variation takes place per a hysteretic cycle denoting energy dissipation whenever the loading magnitude is sufficient to induce a phase transformation. The constitutive model used to describe the shape memory alloy behavior is a modified version of the Brinson model for the one-dimensional case. To provide all thermomechanical properties of shape memory alloy wire, a complete characterization process was performed. Due to numerical reasons, the size of the model of the rotating system was reduced. Finally, numerical and experimental results demonstrate the success of shape memory alloy applied to the suspension of rotating machines as an interesting alternative for vibration control.

Rotors on magnetic bearings rely on external controls to guarantee stability and are designed in case of partial or total failures, when impacts happen and potentially lead to a breakdown. Therefore backup bearings are indispensable. In such rotor-stator interactions the main undesired phenomenon is the backward whirl. The current work investigates the experimental behavior of a horizontal rigid rotor interacting laterally with two types of backup bearings during run up testing. The experimental data is analyzed by orbit analysis, spectrum analyzers, and force magnitudes collected by sensors installed. It is shown experimentally the nonlinear behavior of the rotor-bearing system and the elimination of backward whirl. The advantages and drawbacks of each type of backup bearing are given.

The dynamics of air foil bearings (AFBs) are not yet fully captured by any model. The recent years have, however, seen promising results from nonlinear time domain models, and simultaneously coupled formulations are now available, avoiding the previous requirements for undesirably small time steps and temporal convergence studies. In the present work, an alternative foil structure model is substituted for the simple elastic foundation model to avoid its inherent limitations. The new foil model is based on a truss representation from the literature, but incorporates the foil mass and a dynamic friction model. As a consequence of the friction model's velocity dependency, the foil mass is included to obtain a set of differential equations that can be coupled to the rotor and fluid domains while allowing a simultaneous solution. Considerations leading to a practically applicable implementation are discussed and numerical results are compared with experimental data. The model predicts natural frequencies and mode shapes well, but it is not yet capturing the unbalance response when friction is considered. Possible causes for this discrepancy are discussed and it is suggested that sticking is a more prevalent state than previously assumed.

This paper demonstrates the design and simulation results of model based controllers for AMB systems, subjected to uncertain and changing dynamic seal forces. Specifically, a turbocharger with a hole-pattern seal mounted across the balance piston is considered. The dynamic forces of the seal, which are dependent on the operational conditions, have a significant effect on the overall system dynamics. Furthermore, these forces are considered uncertain. The nominal and the uncertainty representation of the seal model are established using results from conventional modelling approaches, i.e. Computational Fluid Dynamics (CFD) and Bulkflow, and experimental results. Three controllers are synthesized: I) An H∞ controller based on nominal plant representation, II) A μ controller, designed to be robust against uncertainties in the dynamic seal model and III) a Linear Parameter Varying (LPV) controller, designed to provide a unified performance over a large operational speed range using the operational speed as the scheduling parameter. Significant performance improvement is shown for robust control, incorporating model uncertainty, compared to nominal model based control.

This is part I of a twofold paper series, of theoretical and experimental nature, presenting the design and implementation of model-based controllers meant for assisting the hybrid and developing the feedback-controlled lubrication regimes in active tilting-pad journal bearings (active TPJBs). In part I, the flexible rotor-active TPJB modelling is thoroughly covered by establishing the link between the mechanical and hydraulic systems for all regimes. The hybrid lubrication is herein covered in depth; from a control viewpoint, an integral controller to aid such a regime is designed using model-based standard tools. Results show slight improvement on the system dynamic performance by using the hybrid lubrication instead of the passive one. Further improvements are pursued with the active lubrication in part II.

This is part II of a twofold paper series dealing with the design and implementation of model-based controllers meant for assisting the hybrid and developing the feedback-controlled lubrication regimes in active tilting pad journal bearings (active TPJBs). In both papers theoretical and experimental analyses are presented with focus on the reduction of rotor lateral vibration. This part is devoted to synthesising model-based LQG optimal controllers (LQR regulator + Kalman Filter) for the feedback-controlled lubrication and is based upon the mathematical model of the rotor-bearing system derived in part I. Results show further suppression of resonant vibrations when using the feedback-controlled or active lubrication, overweighting the reduction already achieved with hybrid lubrication, thus improving the whole machine dynamic performance.

This work constitutes the first step in a research effort aimed at studying the feasibility of introducing an active lubrication concept in tilting pad journal bearings (TPJBs) that feature the leading edge groove (LEG) lubrication system. The modification of the oil flow into each pad supply groove by means of servovalves renders the bearing active. This article focus on obtaining and validating a numerical model capable of simulating the studied system, in terms of its steady state and dynamics characteristics. The developed model is then used to simulate a simple system, in order to portray the feasibility of affecting its static and dynamic properties by introducing this novel active bearing design.

This work gives a theoretical contribution to the problem of modelling air foil bearings considering large sagging effects in the calculation of the non-linear transient and steady state response of a rigid rotor. This paper consists of two parts: the development of a miltiphysics model of the air foil bearing, and a numerical parameter study of a rigid journal supported in an air foil bearing with a partially supported top foil. The mathematical model of the air foil bearing is centred around the finite element models of both the air film and the top foil structure. These finite element models utilise two types of eight-node isoparametric elements. The rotor is modelled as a rigid body without rotational inertia, i.e. as a journal. The bump foil is included via a bilinear version of the simple elastic foundation model. This paper introduces the bilinear simple elastic foundation model, which combined with the top foil structure model, enables a separation of the top foil and the bump foil. A phenomenon associated within areas of the top foil is where the aerodynamic pressure is sub-ambient. The parameter study investigates the performance of three air foil bearings with partially supported top foils and one air foil bearing with a fully supported top foil. The steady state responses of a journal supported by these air foil bearings are investigated for varied rotational speeds and journal unbalances as well as the top foil sagging in the unsupported area. The study reveals that sub-harmonic vibrations associated with a large journal unbalance can be eliminated by a proper design layout of the bump foil, i.e. placement of the unsupported area. The positive effect is attributed to ‘equivalent shallow pockets’ formed by the sagging top foil.

Railway turnouts are elements exposed to an accelerated track degradation when compared to plain line track. The particular and complex wheel/rail contact characteristics in railway turnouts make the contact forces acting on the wheel/rail interface particularly high. As a consequence of the train/track interaction force magnitude, track degradation is more accelerated here, having a big impact on the general costs used for track maintenance. The contact phenomenon has a stochastic nature due to the great number of variables involved in the interaction problem. Moreover, the uncertainty levels associated with each of the aforementioned variables require in-depth stochastic analyses that reflect the real variability of the parameters which characterize the dynamic interaction and consequently the degradation problem at railway turnouts. The scope of this work is to perform a probabilistic assessment of a railway switch and crossing, aided by the multi-body simulation software, GENSYS. The critical track components with a major impact on track degradation are identified by means of a sensitivity assessment based on the elementary effects method while the uncertainty is assessed by using the Monte Carlo simulation method. The proposed methodology used to evaluate the probabilistic problem is a suitable approach to identify the railpad stiffness, the mass of the bogie, the damping value of the primary suspension and the ballast stiffness, being the critical parameters with the highest impact in the uncertainty analysis of track degradation at turnouts.

Flywheel energy storage systems (FESSs) based on active and passive magnetic bearings (MBs) have a huge energy storing potential in mobile applications. Under such operating conditions the FESS is exposed to gyroscopic forces, which can be handled by mounting the FESS in a passive gimbal. The purpose of this article is to investigate the coupled dynamics among the flexible bodies: gimbal, housing, and rotor in a FESS. This is carried out by theoretical modal analysis of each body and the FESS by utilizing the finite element (FE) method. Modeling of the bodies are carried out in ANSYS MAPDL using SOLID186 elements. The model of the FESS has been built by assembling the FE models of the bodies, representing the MB forces by linear springs using the MATRIX27 element. The first six natural frequencies and mode shapes have been identified for the gimbal, housing, and rotor occurring in the range of 195-801 Hz, 2882-4580 Hz, and 1346-6504 Hz respectively. The model of the FESS indicates 23 natural frequencies in the range 0-2000 Hz. Decoupled vibrations are noticed, such as isolated vibrations corresponding to the fifth mode shape for the gimbal and the first bending mode for the rotor. Coupled vibrations are also observed between the gimbal, housing, and rotor for which the active MB controllers must be designed taking such dynamics into account.

Despite its merits, the Gas Foil Bearing (GFB) suffers from several inherent limitations which could likely be overcome using active radial injection. This hence represents a natural next step, even though several issues relating to the foil structure modelling remain unresolved. A prerequisite for the development of any model-based feedback control scheme is a model capturing the effects of gas injection on the system dynamics. The currently presented work on a generic numerical recipe for GFB simulation is intended as a step towards such a model.The aim of the present work is to consolidate the existing state of the art knowledge on GFB modelling into a generic framework that can act as an efficient platform for further research. By creating a structure where the same pieces of code can be applied for simulation of a wide range of rotors supported by any number of rigid or compliant type gas bearings — with or without injection — the available experimental results can be utilized optimally for validation. Furthermore, the programming intensive optimizations necessary to obtain tolerable solution times can be more easily reused.The framework consists of three domains treating the rotor, the foil structure and the fluid film respectively, along with clear-cut domain interfaces based on linear mappings. In the fluid domain, the film is modelled using the Modified Reynolds Equation dizcretised using finite volume, leaving the injection flow to an auxiliary model. Both the rotor and the compliant structure are represented in generic state-space formats facilitating various different models for both domains. The global system is solved both statically and in time using efficient general purpose routines exploiting e.g. analytical Jacobian matrices and sparse direct linear solvers.

The lack of damping of radial gas bearings leads to high vibration levels of a rotor supported by this type of bearing when crossing resonant areas. This is even more relevant for flexible rotors, as studied in this work. In order to reduce these high vibration levels, an active gas bearing is proposed. The control action of this active bearing is selected based on two different strategies: a simple proportional integral derivative controller and an optimal controller. Both controllers are designed based on a theoretical model previously presented. The dynamics of the flexible rotor are modelled aided by the finite element method and the rotor–fluid interaction in the gas bearing is included using the solution of a modified version of the Reynolds equation for compressible fluids, taking into account the piezoelectrically controlled jet action. Performance and accuracy of both model-based controllers are compared against experimental results, showing good agreement. Theoretical and experimental results show a significant increase in the damping ratio of the system, enabling the flexible rotor to run safely across the critical speeds and up to 12,000 rev/min, i.e. 50 percent over the second critical speed without any instability problems.

Damping factors and natural frequencies of a flexible rotor supported by a gas bearing with piezoelectrically adjusted flow, are theoretically determined using a rotor finite element model coupled with the modified Reynolds equation. An extra term is added to the standard formulation of Reynolds equation aiming at incorporating the effect of the adjustable external pressurized inlet flow. Two different configurations are theoretically as well as experimentally studied: (a) the air is injected from a single orifice positioned at the bottom of the bearing and (b) the air is injected through four radial injectors equally pressurized. For the two configurations, the theoretical results are experimentally validated as a function of the piezoactuators input voltage and the journal angular velocity. Results show a good agreement for natural frequencies and damping factors. Theoretical and experimental results show qualitatively as well as quantitatively that the injectors position and the injection flow (dependent on the piezoactuator input voltage) have an important influence on the dynamic characteristics of the rotor-bearing system. By using one single injector positioned at the bearing bottom, the damping factor associated with the first mode shape can be increased by 10 times when compared to four injectors equally pressurized.

The modified Reynolds equation for active lubrication has been the cornerstone around which the theoretical investigations regarding actively lubricated bearings have evolved over the years. Introduced originally in 1994, it enables to calculate in a simplified manner the bearing pressure field as a function of servovalve controlled pressurized oil injection. This article deals with a preliminary critical review of the simplificatory assumptions that are introduced into the modified Reynolds equation in order to model the phenomena taking place in the interface between the injection nozzle and the bearing clearance. The analysis is performed by means of direct comparison of the results of the modified Reynolds equation model versus benchmark CFD calculations, applied to a geometry representative of the system analyzed. The results show that the modified Reynolds equation mathematical simplicity comes at the cost of reduced accuracy regarding the description of the oil velocity field in the vicinities of the injection nozzle. On the other hand, the modified Reynolds equation results provide sufficient accuracy regarding global magnitudes, such as resulting loads and injected flows, which are essential for designing and operating bearings featuring the active lubrication system.

The work gives a theoretical and experimental contribution to the problem of smart materials connected to double curved flexible shells. In the theoretical part the finite element modeling of a double curved flexible shell with a piezoelectric fiber patch with interdigitated electrodes (IDEs) is presented. The developed element is based on a purely mechanical eight-node isoparametric layered element for a double curved shell, utilizing first-order shear deformation theory. The electromechanical coupling of piezoelectric material is added to all elements, but can also be excluded by setting the piezoelectric material properties to zero. The electrical field applied via the IDEs is aligned with the piezoelectric fibers, and hence the direct d33 piezoelectric constant is utilized for the electromechanical coupling. The dynamic performance of a shell with a microfiber composite (MFC) patch is investigated using frequency response functions (FRFs) obtained via external impact test as well as internal random signal excitation using the MCF patch as an actuator. The experiments are used to validate the numerical results. Good agreement between theory and experiments is obtained in a large frequency range. Discrepancies and insights into optimal modeling frequency range and non-linear behavior are discussed.

The thermo-mechanical behaviour of pseudoelastic shape memory alloy helical springs is of concern discussing stabilised and cyclic responses. Constitutive description of the shape memory alloy is based on the framework developed by Lagoudas and co-workers incorporating two modifications related to hardening and sub-loop functions designated by Bézier curves. The spring model takes into account both bending and torsion of the spring wire, thus representing geometrical non-linearities. Simplified models are explored showing that a single point in the wire cross section is enough to represent the global spring behaviour in spite of complex stress–strain distributions. The experiments are carried out considering different deflection amplitudes, frequencies and ambient temperatures, which influence the spring behaviour to different extents. The model is fitted against a calibration data set resulting in 1.3% residual standard deviation relative to the full range force. Compared to the validation data set, the errors are below 10% relative to the full range of the complex modulus. Uncertainty analysis of the model parameters using a Markov chain Monte Carlo technique shows low to high parameter correlation, and the relative uncertainties are less than ±12%. Both the heat capacity and the convection coefficient are clearly identifiable from the performed experiments.

In this work, the nonlinear dynamic behaviour of a vertical rigid rotor interacting with a flexible foundation by means of two passive magnetic bearings is quantified and evaluated. The quantification is based on theoretical and experimental investigation of the non-uniformity (anisotropy) of the magnetic field and the weak nonlinearity of the magnetic forces. Through mathematical modelling the nonlinear equations of motion are established for describing the shaft and bearing housing lateral dynamics coupled via the nonlinear and non-uniform magnetic forces. The equations of motion are solved in the frequency domain by the methods of Finite Difference and pseudo-arclength continuation. The theoretical findings are validated against experiments carried out using a dedicated test-rig and a special device for characterisation of the magnetic anisotropy. The characterisation of the magnetic anisotropy shows that it can be quantified as magnetic eccentricities having an amplitude and a phase, which result in linear and parametric excitation. The magnetic eccentricities are also determined using the steady-state response of the rotor–bearing system due to forcing from the magnetic anisotropies and several levels of mass imbalance. Discrepancies in the results from the two methods in terms of magnetic eccentricity magnitude are due to additional geometric eccentricities in the shaft. The steady-state system response shows clear nonlinear phenomena, e.g. bent resonance peaks, jump phenomena and nonlinear cross-coupling between the two orthogonal directions, especially during counter-phase motion between shaft and bearings. The clear nonlinear behaviour is facilitated by the lack of damping resulting in relatively large vibrations. The overall nonlinear dynamic behaviour is well captured by the theoretical model, thereby validating the modelling approach.

This paper presents the results obtained from a research work investigating the performance of different Adaptive Neuro-Fuzzy Inference System (ANFIS) 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. Two types of models were developed; the first type uses current accelerations only while the second type considers both the current accelerations and the historical values. 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 ANFIS models can be used to set up reliable force predictors for dynamical loaded flexible structures, 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. The findings showed that the model type employing historical values gives better predictions when operating in their trained regions while the models using only current values have generally higher prediction errors in trained regions but are less sensitive to changes of the system dynamics history.

The dynamic characteristics of rotors supported by Gas Foil Bearings (GFBs) are commonly assessed using linear force coefficients, which are usually derived from an analytical perturbation in combination with an implicit compliance model. This remains common practice even though discrepancies have been reported between the Onset Speed of Instability (OSI) predicted using such linear coefficients and observations from non-linear time integration. For the first time, the present paper pinpoints the root cause of this discrepancy by demonstrating an extended perturbation, akin to that used for tilting pad journal bearings, to predict exactly the same OSIs as obtained from an eigenvalue analysis based on the Jacobian of the full non-linear system of equations. To demonstrate this, the OSIs predicted using (i) the classical perturbation, (ii) the extended perturbation, and (iii) the Jacobian eigenvalues are compared over a range of compliance levels covering essentially rigid bearings to modern heavily loaded industrial GFBs. Using carefully aligned implementations, the deficiency of the classical method is demonstrated to increase with the compliance level (up to 13% for the benchmark case), while the extended perturbation provides OSIs in agreement with the Jacobian eigenvalues for all compliance levels. Furthermore, the mode shapes attainable from the three approaches, encompassing only the rotor, the rotor and foil, and the rotor, foil and pressure respectively, are compared.

The magnetic bearing rotors are present nowadays in many industrial applications. However, a damage or a malfunctioning of the magnetic bearing can produce a dangerous high lateral vibration.Backup bearings hold them to protect the machine integrity. The present study investigates the effectiveness of a pinned bearing on a horizontal rotor compared with a conventional rolling bearing. Experimental tests have been conducted; a mathematical model to describe the motion is then developed and validated numerically. It is shown that both the simulation and the test rig are in good agreement with each other and that the pinned bearing is able to reduce the interval of impacting velocities. The experimental set up is able to gather data fast enough to observe the dynamics and the forces acting on the structure. The orbits and the magnitudes of the impact forces are the key elements to assess the validity of the pinned in contrast to the rolling one as a relevant type of backup bearing.

The papers focuses on the modelling and experimental validation the vibro-impact dynamic behaviour of rotors interacting with two types of backup bearings, i.e. one pinned backup bearing with polymeric pins (original contribution to the problem) and another common ball bearing (conventional). The impact forces are modelled using Hunt and Crossley approach. The parameters of the constitutive equation responsible for describing the contact forces are a priori identified in auxiliary tests. The vibro-impact model is built by coupling the nonlinear contact forces with the rotor-bearing system dynamics and the theoretical results are obtained by integrating the coupled equations in time. A fully-instrumented test bench is designed, built and used to validate the experimental results. The effectiveness of the pinned bearing is evaluated in terms of orbits and maximum vibration.

Proper dimensioning of magnetic bearings for non-static gimballed FESS is currently hindered by the lack of models that can predict the maximum forces in the bearings. If FESS is to compete with conventional electro-chemical batteries in terms of energy density, the magnetic bearings must be dimensioned optimally for weight and size. Magnetically suspended FESS has been experimentally investigated; however, the methods for theoretically predicting magnetic bearing loads are still limited. This paper presents how to determine such magnetic bearing loads in a gimballed FESS subject to a moving foundation. The predicted forces are compared with measurements from an experimental test bench with good agreement. Furthermore, the maximum forces in a gimballed FESS are compared with maximum forces in a non-gimballed FESS. It was found that the gyroscopic forces in the non-gimballed case quickly become critical making the gimbal mount essential for reducing magnetic bearing loads.

The Gas Foil Bearing (GFB) is a promising and environmentally friendly technology allowing support of high-speed rotating machinery with low power loss and without oil or electronics. Unfortunately, GFBs provide limited damping, making an accurate prediction of the Onset Speed of Instability (OSI) critical. This has traditionally been assessed using linearised coefficients derived from the perturbed Reynolds Equation with compliance included implicitly. Recent work has, however, revealed significant discrepancies between OSIs predicted using these techniques and those observed from nonlinear analysis. In the present work, the perturbation method’s underlying assumption on the pressure field is investigated by including the hitherto neglected pressure–compliance dependency directly. This leads to an extended perturbation akin to that commonly applied to tilting pad bearings and is shown to predict OSIs with much better agreement to time integration results. The extended perturbation method is cumbersome, but serves to highlight the error introduced when applying the classical perturbation method—as developed for rigid bearings by J. W. Lund—to GFBs.

Gas bearing systems have extremely small damping properties. Feedback control is thus employed to increase the damping of gas bearings. Such a feedback loop correlates the input with the measurement noise which in turn makes the assumptions for direct identification invalid. The originality of this article lies in the investigation of the impact of using different identification methods to identify a rotor-bearing systems’ dynamic model when a feedback loop is active. Two different identification methods are employed. The first method is open loop Prediction Error Method, while the other method is the modified Hansen scheme. Identification based on the modified Hansen scheme is conducted by identifying the Youla deviation system using subspace identification. Identification of the Youla deviation system is based on the Youla–Jabr–Bongiorno–Kucera parametrisation of plant and controller. By using the modified Hansen scheme, identification based on standard subspace identification methods can be used to identify the Youla deviation system of the gas bearing. This procedure ensures the input to the Youla deviation system, and the noise is uncorrelated even though the system is subject to feedback control. The effect of identifying the Youla deviation system compared to direct subspace identification of the gas bearing is further investigated through a simulation example. Experiments are conducted on the piezoelectrically controlled radial gas bearing. A dynamic model is identified using the modified Hansen scheme as well as using Prediction Error Method identification. The resulting models are compared for different imperfect nominal models, to examine under which conditions each method should be used.