ABSTRACT. This paper defines methodology to perform modal and frequency response analysis of a structural system composed of static parts and different rotors (i.e. whole engine model) subjected to rotor non-axisymmetric loading. One example of non-axisymmetric loading is a non-axisymmetric inlet pressure on an engine fan in a cross-wind environment. Although this load is static from a stationary observer point of view, each fan blade, due to its rotation, is seeing an alternated pressure that is promoting a dynamic excitation. This type of loading is designated as 0EO (zero engine order) in a stationary frame. Other example of non-axisymmetric static load is the gravity load: again, a rotor blade is seeing that this load is an alternating load due to blade rotation. RotRed methodology is used to perform rotor model reduction from a rotor cyclic symmetry 3D model. This methodology incoporates all rotating effects (gyroscopic effect, stress stiffening and spin softening) in stationary system, that allows incorporation of other rotors at different speeds and static parts in the same whole engine model. Normally, excitation type considered in whole engine model is out of balance on each rotor that corresponds to a 1EO excitation type. In this paper, it is shown how RotRed reduced rotor model matrices can be used to perform 0EO modal analysis and frequency response of a complete whole engine model for this non-axisymmetric static load type. In a stationary system Campbell diagram, 0EO excitation line corresponds to horizontal line axis. Whole engine models that have backward modes crossing horizontal line will have resonances under 0EO excitation type. This methodology can be used to obtain dynamic response (displacements and stresses) of a whole engine model when subjected to non-axisymmetric loading (i.e. non axisymmetric inlet pressure in fans), taking into account dynamic amplification factor if resonances are within running range or if maximum speed is approaching first resonance.

ABSTRACT. Thrust collars are an established machine element in integral gear units to transmit axial forces between wheel and pinion shaft(s).

In this paper it is shown that the design of the thrust collar – especially the clearance – can cause a resonance phenomenon, at which the axial and/or yaw motion of the gear wheel is coupled with the lateral motion of a pinion shaft. An excitation of this specific motion can for example be induced by an axial run-out of the thrust collars contact face. As a result, unwanted lateral vibrations are observed, at first in the mechanical running test of the gear unit.

For the investigations in this paper, a multi-body simulation model is used. In addition to the simulation, a phenomenological comparison with results from the mechanical running test is made.

ABSTRACT. Gear trains are plagued by self-excited vibrations that are concentrated at the mesh frequency and its harmonics due to their varying mesh stiffness and deviations from the ideal involute profile. This is even more pronounced in spur gears due to their lower contact ratio in comparison to helical gears. For an electric vehicle, due to the absence of an internal combustion engine, noise and vibration signature of the gearbox becomes an important aspect of the vehicle's comfort. However, the presence of a traction motor offers the advantage of having a potential actuator for actively countering these vibrations without adding any additional weight or packaging constraints. This paper presents a fundamental insight into the effect of introducing controlled torque ripples at the mesh frequency and its harmonics, on the noise and vibration characteristics, and the efficiency of the gear mesh. The study utilises a dynamic model of a single stage gear train that accounts for the time varying mesh stiffness and sliding friction at the gear teeth contact. This model is used to provide an understanding of gear mesh dynamics and their resulting interaction with the imposed torque ripples. The study demonstrates the positive effects that controlled torque ripples can have on the noise and vibration behaviour of gear trains and the underlying mechanics that govern this improvement.

ABSTRACT. The paper describes a coupled torsional-bending vibration model of a ship's propulsion shaft system with a residual shaft bow. The developed model presents an extension of the Jeffcott rotor model. In order to test the numerical model and determine the coupled torsional-bending vibrations, several cases were analyzed. First, a reference case corresponding to a fully axisymmetric ship propulsion system is set up. Then a shaft bow was introduced, and the influence of the constant radial force in the vertical direction at the position of the stern bearing was analyzed. Such conditions exist when a ship navigates on a calm sea under a partial or fully loaded hull. Finally, the case of sailing on a rough sea is analyzed when the propeller racing occurs due to the stern lifting out of the sea. Based on the results of numerical analysis, it was found that the proposed model well describes the coupling of torsional and bending vibrations of the propulsion shaft system with residual shaft bow. The aim of the research is to prepare input data for estimating the fatigue life of the propulsion shafting system of ships sailing at slow steaming speed for low carbon shipping.

ABSTRACT. Current methods of rotor system condition monitoring require a substantial amount of manual work for large fleets of machines. Data-driven automated fault diagnosis models based on deep learning (DL) have the potential to drastically reduce the amount of time spent on manual analysis. However, there is often a lack of data from the entire range of possible operating conditions of a system. The poor generalisation of most DL based models to operating conditions not present in the training data decreases their usefulness in industry applications. This paper investigates the generalisation ability of a few-shot learning method using only a single example to learn each new class. A prototypical network with a modified Deep Convolutional Neural Networks with Wide First-layer Kernels (WDCNN) architecture was used as the few-shot model. The generalisation of the model was studied from sensor to sensor and across operating speeds. The results indicate that the model is robust to changes in sensor orientation and relatively robust against changes in sensor location. Additionally, the model showed promising results when tested on operating speeds many times higher or lower than it was trained on. The results show that few-shot learning methods have the potential to work in industry applications where limited training data is available. This research also gives an excellent baseline for future research on the generalisation of few-shot learning methods on rotor system fault diagnosis over large changes in operating conditions.

ABSTRACT. Identification of lubricated bearings parameters represents an activity of great interest in real applications, either to allow the adjustment of numerical models, to verify and correct operating conditions, or even to use these parameters for monitoring the equipment. Over the years, several methods for identifying dynamic coefficients of lubricated bearings have been used by the scientific community. Among these methods, filtering techniques have shown to be promising in different applications, since they present a relatively low computational cost and consider prediction-correction strategies. Thus, this work aims to evaluate the use of the extended Kalman filter (EKF) technique in the estimation of the roller bearing total parameters. Numerical tests performed in this work showed that EKF can successfully estimate roller bearing parameters, although low damping values can be insensitive to the dynamic response of the contact and, consequently, cause deviations in the final estimation. The results obtained in this work have shown satisfactory parameter estimation considering different signal-to-noise ratios for the measured signal.

ABSTRACT. One of the most important issues that engineers have always been concerned of is the study of oscillations and the stability of a structure. Rotating systems and especially shafts exist in a wide range of applications, such as motors, turbines, compressors, pumps and more. In these systems, an uncontrolled oscillation can lead to resonance, which may result in devastating consequences for the machine, but also for human safety. So, it has been made clear that it is important to analyse the behaviour of rotating systems, in many situations, so to predict an error and especially avoid an accident. In this study a fault diagnosis algorithm was developed for roller bearings. For this reason, experimental data were used to train a model in MATLAB environment. This model can diagnose a fault in roller bearings with the usage of the SVM method. Firstly, the signal is divided into eight datasets. The model is trained using different number of datasets. Then predictions were made with the rest of the data, so to check how the algorithm works with new signal but also to understand how the amount of data that wereused to train the model, affects its effectiveness. In conclusion, a success rate of 95.83% is achieved for the models that had been trained with the 2/8 datasets, 79.17% for the models that had been trained with the 4/8 datasets and 100% for the models that had been trained with the 6/8 datasets. At this point, it is worth mentioning that during the training of the model, both healthy and faulty parts of the signal were used.

ABSTRACT. Variation in powertrain parameters caused by dimensioning, manufacturing and assembly inaccuracies may prevent model-based virtual sensors from representing physical powertrains accurately. Data-driven virtual sensors employing machine learning models offer a solution for including variations in the powertrain parameters. These variations can be efficiently included in the training of the virtual sensor through simulation. The trained model can then be theoretically applied to real systems via transfer learning. This research presents a training procedure for a data-driven virtual sensor. The virtual sensor was made for a powertrain consisting of multiple shafts, couplings and gears. The training procedure generalises the virtual sensor for a single powertrain with variations corresponding to the aforementioned inaccuracies. The training procedure includes parameter randomization and random excitation. That is, the data-driven virtual sensor was trained using data from multiple different powertrain instances, representing roughly the same powertrain. The virtual sensor trained using multiple instances of a simulated powertrain was accurate at estimating rotating speeds and torque of the loaded shaft of multiple simulated test powertrains. The estimates were computed from the rotating speeds and torque at the motor shaft of the powertrain. This research gives excellent grounds for further studies towards simulation-to-reality transfer learning, in which a virtual sensor is trained with simulated data and then applied to a real system

ABSTRACT. The increasing demand for high-speed rotor-bearing systems results in the application of complex materials, which allow for a better control of the vibrational characteristics. This paper presents a model of a rotor including viscoelastic materials and valid up to high spin speeds. Regarding the destabilization of rotor-bearing systems, two main effects have to be investigated, which are strongly related to the associated internal and external damping of the rotor. For this reason, the internal material damping is modeled using fractional time derivatives, which can represent a large class of viscoelastic materials over a wide frequency range. In this paper, the Numerical Assembly Technique (NAT) is extended for the rotating viscoelastic Timoshenko beam with fractional derivative damping. An efficient and accurate simulation of the proposed rotor-bearing model is achieved. Several numerical examples are presented and the influence of internal damping on the rotor-bearing system is investigated and compared to classical damping models.

ABSTRACT. As torsional measurement technologies advance, machine trains may have different modalities measuring torsional vibration. For example, a strain/stress measurement reports torsional stress at a coupling and a demodulated signal from a toothed wheel at the end of reports torsional speed variation. How does the rotordynamicist or diagnostic engineer reconcile these measurements?

This paper begins with a review of common torsional vibration measurement technologies (encoders, tachometers, strain, and magnetostrictive). The review includes a first principles explanation of how the various instruments measure torsional vibration, a discussion on practical application to turbomachinery, and relative advantages and disadvantages of each measurement modality.

Next the paper uses state space models with lumped parameter and finite element model parameters to build a model incorporating both the mechanical elements and the various measurement approaches. Simulations using this hybrid mechanical/instrumentation model show responses from the different modalities as well as provide a transfer function that relates the reported values.

This model is adapted to a test kit with a known torsional forcing function and multiple measurements to show how to relate the transfer function to a physical machine with sensors. Finally, the paper then applies these principles to a physical gas turbine in power generation equipped with multiple torsional measurement systems.

The paper concludes with guidelines for applying these principles to field data. The guidelines include filtering and component extraction techniques, velocity to displacement conversion, and guidelines for state-space model simplification.

ABSTRACT. Over the wild range of geared systems, the planetary gear sets are distinguished by their capacity to provide high gear ratio in a compact package. They are used for example in automatic gearboxes, transmissions for hybrid vehicles, energy production systems such as wind turbines, home automation applications such as shutters and blinds. However, contrary to cylindrical gears with fixed and parallel axes, predicting and controlling the whining noise remains a difficult problem because of the coupling between the multiple gear meshes and the mobility of the planets axes.

It is well know that the gear whining noise is generated by the mesh process. The problem posed is multi-scale in nature. Indeed, the overall dynamic and vibroacoustic behaviour of geared systems (on the scale of a meter) depend on the local micro-geometry of the teeth (of the scale of a micron), associated with the transmission error. Moreover, the problem is parametric in nature, due to the periodic fluctuation of the mesh stiffness, and non-linear, due to the presence of functional clearance and close contacts between teeth and the bearings. These parametric internal excitations generate dynamic mesh forces which are transmitted to the housing through wheel bodies, shafts and bearings. The housing vibratory state is directly related to the radiated noise.

In the case of planetary gear sets, predicting the housing vibratory state presents a high challenge. The carrier rotation modulates the housing vibration response, at its rotational frequency. The iterative spectral method allows the solving of linear parametric equations of motion, in the carrier reference frame, in the spectral domain, with short computational time. The dynamic response at meshes is hence fully characterised. However, the computation of the dynamic response of any fixed point on the ring gear requires an additional step. An original approach is proposed, by taking into account the modulation effects induced by the relative rotation between the observation point (fixed point located on the ring gear) and the meshes (attached to the carrier reference frame). Finally, a comparison with experimental measurement is presented.

ABSTRACT. This paper investigates the torsional response of an electric powertrain considering the effect of the control methods of the electric motor drive, consisting of a variable frequency drive and an electric motor. The vibratory torque occurring in the mechanics of an electric powertrain driven with classical induction motor control methods (open- loop V/Hz and closed-loop V/Hz) were compared. The simulated powertrain was accelerated to a chosen operating speed and a sinusoidal torque excitation was applied. The shaft vibratory torque was calculated over a range of motor operating speeds. The analysis was conducted on simulation results produced using an electric motor drive simulator and a torsional vibration analysis software. The simulation results indicate that the control has a major effect on the torsional response of the powertrain, especially in the case of an improperly configured control system. The combination of the software used is an open-source toolchain for simulating the effects of motor control on the torsional vibration response of powertrain mechanics.

ABSTRACT. Electrical machines are inherently capable of developing transient torque considerably in excess of rated value when exposed to a momentary voltage interruption and reclosing. The magnitude of this transient torque may reach to 20 times rated torque. The largest electrical air-gap torque between the rotor and stator can be calculated for the worst out-of-phase reclosing. However, the transmission of this airgap torque to the mechanical powertrain components depend on the stiffness and inertia characteristics of the system. To define a dimensioning torque for each component, a traditional method has been to calculate first the air-gap torque and apply thereafter this torque to a mechanical drivetrain model as loading. However, this approach neglects totally the interaction between the electrical and mechanical systems. The aim of this paper is to introduce a novel method to determine the maximum torque of flexible powertrain components during the breaker reclosure with the most unfavorable conditions. The applied approach is based on a coupled electromechanical simulation of an electrical motor, driven machine and flexible drivetrain. The approach is based on the well-known models of electromagnetics and mechanical drivetrain. The novelty of this paper is the strong coupling between these models and the application of this combined model to the reclosure analysis. It can be added that a non-linear model of electromagnetics is needed because, for example, the supply voltage drops to zero and rises back again. As a numerical example, a reclosure analysis of a motor-reciprocating compressor with a flywheel and a flexible coupling is used. The worst reclosing conditions are identified and the maximum torque determined. The presented approach seems to fit well for the dimensioning of powertrain components.

ABSTRACT. Majority of industrial machinery with main working tools characterized by rotational motion, i.e. the rotor machines, are driven by electric motors. Such electro-mechanical systems are often affected by detrimental torsional vibrations, suppression of which is commonly performed by more or less effective passive, semi active and active dampers. In this paper there is proposed an alternative method of attenuation of torsional vibrations in such objects. Namely, an asynchronous motor under the proper control can simultaneously operate as a source of drive and actuator. Here, rotational velocity oscillations of the motor rotor superimposed on its average rotational speed result in additional fluctuations of electric motor currents which generate oscillating components of the asynchronous motor torque characterized by the same frequency as this of torsional vibrations affecting the entire drive system. By means of a proper control of the motor operation it is possible to precisely counterweigh the external excitation by the electromagnetic motor torque in order to suppress torsional vibrations in the object under study. Using this approach, transient and steady-state torsional vibrations of the rotor machine drive system can be effectively attenuated as well as its precise operational motions can be assured. The theoretical investigations are carried out by means of a structural mechanical model of the drive system and an advanced circuit model of the asynchronous motor controlled using two methods: the direct torque control DTC-SVM as well as using a simplified, but effective and robust approach based on the motor voltage supply frequency dependent on the momentary rotational velocity of the driven machine working tool. Numerical calculations have been performed for three typical electromechanical drive systems of the common structure but with different powers, total mass moments of inertia and nominal rotational speeds. These are: the drive systems of the high-speed beater mill and heavy industrial blower as well as the laboratory drive train designed to an experimental verification of the theoretical findings. From the obtained results it follows that for majority of tested cases by means of the abovementioned simplified frequency control technique transient torsional vibrations excited by step-wisely disturbed loadings and steady-state torsional vibrations induced harmonically can be suppressed as effectively as using the advanced vector method DTC-SVM.

ABSTRACT. A test rig was built to perform fatigue tests on thick-walled cylinders made of fibre reinforced plastic (FRP). During the fatigue test, the rotational speed of an FRP cylinder is periodically varied until it fails. The FRP cylinder is connected to a drive spindle that accelerates and decelerates it using a permanent magnet synchronous machine (PMSM). To avoid excessive wear, the rotor is supported by active magnetic bearings (AMB). After the fatigue test was finished with the first cylinder, a new cylinder was attached to the test stand. With this new specimen, previously non-existent vibrations occurred during fast accelerations and decelerations. For high accelerations, these vibrations led to instability of the rotor. However, high accelerations are desirable to perform the fatigue tests in the shortest possible time. Hence, the AMB control should be made insensitive to these vibrations. Since the vibrations depend on the acceleration of the rotor, it is reasonable to assume that they are induced by the PMSM. To reduce the vibrations, these excitations from the PMSM are included in the model-based controller parametrization process for the radial AMB, in which the parameters are adjusted via an optimization process. With the adjusted control, the amplitude of the vibration was significantly reduced and higher accelerations were possible.

ABSTRACT. The rotors of turbomolecular pumps (TMP) are subject to continuous further development in various aspects to meet the increasing requirements. Vibrational and acoustic emissions are becoming more relevant. Applications such as electron microscopes or ion mobility spectrometers require a low vibration level of the TMP used. The rotor of a TMP always has a certain residual unbalance, which results in radial forces that are transmitted to the housing via the bearing and lead to vibrations and noise. In practice the installation of a rotor with a high balance quality does not always result in low vibration and noise emissions from the TMP. The bearing of the TMP rotor consists of a combination of ball bearing and permanent magnet bearing (PMB). The PMB of the TMP with hybrid-bearing is focus of this study. Due to intrinsic imperfections in the magnetic rings, there is an additional radial force that can affect the running characteristics of rotor and pump, called magnetic bearing error. During operation of the TMP, the heating of the rotor leads to an axial displacement between the rotor and stator magnetic rings and thus to a change of the radial rotor forces. With the aid of an experimental investigation, this behavior is reproduced and the effect, that the magnetic bearing error exhibits a strong dependence on the axial displacement between the stator and rotor, is observed for the first time. The experimental results show that this dependence of the magnetic bearing error strongly influences the first resonance of the rotor in terms of amplitude and resonance frequency. With the help of a simulation, a deeper understanding of the magnetization properties and their influence on the generation of additional forces in the rotor is derived. An optimization approach is developed, which enables the targeted assembly of the PMB based on the individual magnetization properties of the individual magnet rings. This reduces the dependence of the magnetic bearing error on axial forces. The derived measures contribute to the overall objective of reducing vibrational and acoustic emissions of TMP and improving the transfer of the balance quality of the rotor to the pump.

ABSTRACT. Several industrial sectors, including aerospace, energy, and automotive, are dependent on rotating machines, such as turbines, motors, turbo generators, pumps, or compressors. Thus, there is a continually growing need to improve performance, whilst reducing costs and pollutant agents associated with system operation. To achieve these goals, strategies to control the rotor dynamic behaviour are critical. One approach to control the dynamics of a rotating system is through changing the bearing characteristics, directly altering the stiffness and/or damping of the system as a whole. A traditional bearing can become an active bearing through combining it with any element that enables control, thus making it an active hybrid bearing. In the case of a traditional conical bearing, its dynamic characteristics could easily be manipulated through compensation of the shaft axial position, which could be achieved via a thrust magnetic bearing. Therefore, this paper proposes to create an alternative hybrid bearing which consists of a radial conical fluid bearing and a thrust magnetic bearing to improve performance. The assessment of the concept is evaluated through the analysis of the rotor in the frequency domain, with results showing that the rotor vibration can be reduced by up to 70%.

ABSTRACT. Typically, the dynamic/mechanical analysis of turbomachinery systems is divided into rotor dynamics, which focuses on the shaft-bearing system with simplified disks, and blade dynamics, which deals with isolated bladed disks. This division into two sections significantly simplifies the modelling and calculation effort. It is permissible if the natural frequencies of the two subsystems are sufficiently separated, in this case the mutual influence is negligibly small. Recent trends towards steam turbines with longer blades with lower natural frequencies that might be close to subcritical natural frequencies of the shaft increase the importance of a combined analysis of both subsystems. In [1] measurement results were presented which show increased blade amplitudes at a certain frequency, which was not predicted by common numerical simulations of the isolated bladed disk. When the rotating speed reaches half of the nodal diameter 1 resonance frequency the blades show increased amplitudes. At the same frequency, a significant dip of the shaft vibrations was observed at the bearings, which implies an interrelationship between rotor and blade vibrations. The authors of [1] were able to reproduce these rotor unbalance induced blade vibrations with a numerical simulation of a coupled rotor-blade-model. As a next step, the exact circumstances and conditions, which are responsible for the described behaviour, must be determined. In this paper, further investigations on the aforementioned industrial rotor-blade-model are carried out to gain further insights on the effects of rotor-blade-coupling. The system is used as a starting point to derive the relevant parameters for the observed rotor-blade-interaction. The findings are then applied to a simplified rotor-blade-model to validate their impact on the system behaviour.

[1] Grein, R., Ehehalt, .U, Siewert, .C, and Kill, N., 2021. "Rotor-Blade Interaction During Blade Resonance Drive-Through." In Proceedings of the ASME Turbo Expo 2021: Turbomachinery Technical Conference and Exposition. Volume 9B: Structures and Dynamics.

ABSTRACT. Turbine blade vibrations are often measured during operation for monitoring purposes as well as during the design development and for basic research and validation. Due to a high mechanical load, these vibrations can lead to damage for example caused by high cycle fatigue. In this paper the dynamic behaviour of an academic free-standing bladed disk is investigated at various operating points using both strain gauges applied on nearly every blade root and supplemental tip timing measurements. The blades first bending mode was tested both linearly and with two different types of nonlinear underplatform friction dampers at two engine orders. The rotation test rig allows experiments under vacuum conditions, so that an aerodynamic excitation of the turbine blades is excluded and only a single engine order is present in each resonance. In order to evaluate the equivalence of the results from both measurement systems, several excitation forces and speed gradients for transient resonance passages were investigated. In addition to the equipment effort, this comparison also includes the assessment of the recorded measurement data. For this purpose, the individual stress-deflection ratios for each blade and their development along the resonance peaks and captured frequency response functions are examined at the considered operating points. During all experiments each individual blade has shown slightly different dynamic properties, which refers to the mistuning effect. In this paper, we considered geometric mistuning, which can change the behaviour of the individual turbine blades to the extent that both resonance amplitude and frequency can vary noticeably, so that the classic Finite Element approach of an ideal blisk cannot represent this span adequately. Therefore, the experimentally determined linear eigenfrequencies of each blade were compared with the results of simulated blade alone frequencies, obtained from mesh-morphed blade FE models. It is shown that both measurement systems provide comparable results for engine runs with friction nonlinearities due to the underplatform dampers, along with very good accuracy during linear testing. Likewise, the morphed blade models are able to represent the trend of the measured mistuning pattern adequately.

ABSTRACT. To describe the dynamics of rotating bladed disk assemblies, so-called mistuning effects must necessarily be taken into account since individual blade vibration amplitudes may exceed the nominally tuned case significantly. Beside material inhomogeneities and variances with respect to material and excitation forces, one of the key sources for mistuning are manufacturing tolerances and, as a consequence, unavoidable geometric uncertainties. Due to the geometric mistuning, the turbine blades will show different eigenfrequencies, which may affect the safety and reliability of the whole aircraft engine or steam or gas turbine. Within this work, the identification of the blade geometry mistuning and its effect on the blade eigenfrequency distribution is investigated. The geometry of an academic blisk with 40 blades is captured by a blue light fringe projection, an optical measurement procedure. Therefore, the measured blisk is airbrushed with a thin Ti2O-layer to obtain a surface, that is as diffusely reflective as possible. The surface scan itself is carried out by means of many perspectives in order to capture almost the whole surface of the blisk. To ensure a high assembling accuracy of the different scan-perspectives, optical reference marks are used, whose 3d-position is captured by a preceding photogrammetry measurement. The deviation from nominal geometry is represented by principal components analysis (PCA), in which the geometric deviation can be represented by several uncorrelated geometric mistuning modes (GMM). In addition to geometry measurement, a mesh-morphing procedure is carried out to generate Finite-Element-meshes for each as-is individual blade. The frequency mistuning pattern of the first bending mode (1F) is calculated for morphed blade models using Finite Element Analysis (FEA), serving for both numerical studies of the mistuned blisk as wells as a comparison with measurement data, gained from rotational experiments using tip timing and strain gauge data.

ABSTRACT. Aircraft engines are usually dual-spool rotor/bearing/casing systems, where the ro-tors may rotate either in the same or in the opposite direction. They are supported by rolling element bearings, which are usually equipped with squeeze film dampers (SFDs). These nonlinear elements are the main source of external damping applied on the rotor system. The need for robust whole engine dynamics under small blade tip clearances, even in cases of higher unbalance, makes the detailed consideration of SFDs in dynamic simulations necessary. Squeeze film dampers as well as hydrodynamic journal bearings depict strong non-linearities in the impedance forces. However, they are usually included in analytical dynamic models of jet engines as linear elements, especially in frequency domain, using linearized stiffness, damping and mass coefficients. In this manner, linear harmonic analysis is feasible and the unbalance response of the system can be evaluated with relatively low computational cost. Contrary to journal bearings, the linearization of SFDs in aircraft engines is not trivial, since the rotor is not whirling in small-radius orbits around a fixed equilibrium. This linearization issue was addressed only recently, despite the fact that SFDs are used in aircraft engines for more than 40 years. The present paper incorporates the linearization of SFDs in rotor dynamic simulations through a harmonic balance approach. The method used is in principle a general approach and can be used on any SFD type or model, e.g. including or excluding piston rings. Moreover, the approach can be extended to other types of nonlinear bearings, like rolling element bearings, following all the advantages and limitations of the harmonic balance method. The method is firstly applied to a Jeffcott rotor mounted on two SFDs in order to provide insights about its accuracy and efficiency, using as a reference a time-transient simulation using nonlinear SFDs. It is shown that, depending on the rotor/bearing system configuration, higher harmonics of the response should be also calculated in order to accurately capture the response of the rotor. Finally, the method is applied to a realistic aircraft rotor/bearing system, where various unbalance and bearing configurations are implemented. The method aims to of-fer a solution on the implementation of load dependent (further to speed dependent) SFD models in highly-efficient and highly-accurate frequency response analysis.

ABSTRACT. The balancing of flexible rotor-bearing systems throughout multiple critical speeds is one of the most challenging tasks in rotor dynamics. Since modal balancing techniques are limited to lightly damped systems, influence coefficient methods or combinations of both methods are frequently used. The disadvantage of conventional influence coefficient balancing methods is that they require multiple trial runs, which are costly and time-consuming. Therefore, modern balancing methods focus on substituting the measurement of influence coefficients with simulations.

Recently, flexible rotor balancing methods based on the Numerical Assembly Technique (NAT) have been proposed. First, a modal balancing method using NAT has been developed, which is only applicable to isotropic rotors supported on roller bearings. Later, a method utilising influence coefficients calculated with NAT has been presented, which is also capable of balancing anisotropic rotor-bearing systems supported on fluid film bearings. The advantages of NAT are that it leads to quasi-analytical solutions and is very computationally efficient. Previous practical tests have shown that the inclusion of internal damping and the foundation influence is likely to improve the balancing accuracy.

The aim of this work is to validate the suitability of an extended form of NAT based on the Timoshenko beam theory for the field balancing of flexible rotors. Therefore, a NAT simulation is used to balance the first two modes of a testbed, consisting of an axial symmetric rotor running on anisotropic supports, without trial runs. In the test configuration, multiple disks of different weights are mounted on the flexible shaft and unbalance weights are applied. The version of NAT used for this application includes the effects of multiple disks, a stepped shaft, anisotropic bearings, internal and external damping, shear deformation, rotary inertia, gyroscopic effects and the foundation influence. The internal damping is included with a viscoelastic material model using fractional time derivatives, which is able to accurately represent a broad class of materials. The mode shapes, eigenvalues and unbalance responses are measured and compared to values calculated with NAT, to show the accuracy of the simulation. The system is successfully balanced using influence coefficients calculated with NAT and a significant reduction of the vibration amplitude is determined.

ABSTRACT. The first part of the article (SIRM 2021) presented educational model of fluid induced instabilities in rotating machinery and its application for solving rotordynamic problems in the industry. Another group of subsynchronous vibration phenomena in this type of machinery are forced parametric vibrations caused by re-excitation of rotor natural frequency due to periodic changes in the rotor support stiffness. Vibrations of this type were in some cases wrongly diagnosed as fluid induced instabilities causing unnecessary costs. It is important to provide simple physical models to give diagnosticians in the industry tools to correctly identify the source of vibration, and such an educational model, based on [1], is presented and its application illustrated by forced subsynchronous vibration cases of rub and of looseness in the support. Sample diagnostic cases include: - generation of exactly ½ X response by rub in the incorrectly machine bearing of 30MW steam turbine generator - generation of exactly ¼ X response by looseness in the bearing support of 200MW combined cycle turbine-generator train Following the standard examples, it is shown how the model can be used for finding the root cause of less usual case, the torsional vibration of the 500MW single shaft combined cycle unit. The problem was identified to be result of relatively small variations of the fuel flow to gas turbine, caused by one of the auxiliary systems provoking the re-excitation of the first torsional mode for the coupled rotors, due to nonlinear stiffness of the diaphragm coupling between the generator and the steam turbine. Considering low amplitude of exciting torque and the high level of vibration response this can be qualified as the example of self-excited vibration phenomenon