This paper considers the challenge of calculating accurate ball bearing stiffness, which is attributed to the uncertainty in the rolling element positions; and it solves that issue based on the simultaneous consideration of these positions and the physical effects of the rotating components. The novelty of the paper consists of the suggested methodology to resolve the uncertainty of the rolling element circumferential position when calculating bearing stiffness. The problem is solved through various formulations of dry and lubricated contact and validated based on the consideration of the finite-element model. The algorithms presented in the paper allow for the calculation of the resulting stiffness based on the stiffness values evaluated through different ball bearing positions. The approach presented in the paper is validated based on the experimental data. For this purpose, the model of the rotor in ball bearings is built, along with further calculation of the rotor dynamics. The comparison of the critical speeds calculated for the rotor in bearings, which have stiffness evaluated by the proposed approach with the measurements for the real machine, indicates a high accuracy of the suggested method in comparison with the methods that consider the single position of the rolling element when estimating their stiffness. In the paper, further recommendations for the use of the presented method are given, which will be useful for engineers in the field of turbomachinery dynamics.

This paper presents an approach to determine the stability of the rotor in herringbone groove gas journal bearings (HGGJB) based on nonlinear transient analysis. The approach considers rotating (on a rotor) or stationary (on a bearing shell) grooves. The bearing gas film model is described by the Reynolds equations. The finite element (FE) method has been applied to obtain an accurate solution for the HGGJB design. The rotor model is discretized with Timoshenko beam finite elements. A study on the stability of a rotary machine supported by HGGJBs is performed to investigate threshold rotating speeds for the cases of stationary and rotating grooves applying a nonlinear transient analysis procedure. As a result, the influence of rotating grooves on system stability improvement has been investigated. Additionally, the correspondence of the calculated results to the experimental data available in publications and the nonlinear effects presented on Waterfall plots are also discussed in the article. The presented method is an extension of previously developed techniques that engineers can use for design and calculation purposes. The nonlinear approach allows accurate simulation of coupled rotor-HGGJB systems and has no limitation for designs with a small number of grooves, as the FE method is used for bearings discretization. The method allows the physical effects of rotating/stationary grooves to be accounted for.

As the plant capacity is increased, the equipment in the plant also needs to be upgraded. Replacing with a new machine is not always the right economical decision when alternative options for upgrading and retrofitting the exiting machine exist. Upgrading the machine not only increases the performance thus reducing the power consumption, but it also increases the system reliability, reducing the downtime and thus helping the plant operate more efficiently. The compressor used for this work was in operation, and due to prolonged operation, the impellers had degraded primarily due to fouling caused by process contaminants. This multi-stage centrifugal compressor was subsequently upgraded to increase the inlet volume flow rate by 20% with the objective of retaining the same casing and rotor shaft. The redesigned compressor stages has better performance with reduced power consumption. The redesigned four-stage centrifugal compressor operating at design speed of ~8856 rpm may be prone to instabilities caused by aerodynamic cross-coupling and hydrodynamic bearing modification. The authors in the present study are exploring any instabilities in the rotor-bearing system that may be caused due to the upgrade of the compressor for higher volume flow rate. To meet the new operational requirements, the bearings were redesigned. The stability analysis was performed considering the destabilizing effects of the aerodynamic excitations, based on which modifications were performed on the design to stabilize the multi-stage centrifugal compressor and to conform to satisfy the API Level I stability criteria. The modifications performed on the rotor/bearing system resulted in reduced vibration amplitudes for the required level of rotor unbalance conditions. These modifications also significantly eliminated the instability that was initially observed, and the redesigned compressor not only met the requirements for the change in operating conditions but also fully complied with API 617 standards.

Hydrostatic bearings are widely used in industry, including aerospace and energy sectors. Hydrodynamic lubrication mechanism has been well studied analytically and experimentally and various types of bearings were developed to provide increasing operating speed, load capacity, stability and efficiency for modern rotating machines. Hydrostatically lubricated bearings have principal difference (in comparison with hydrodynamic bearings) and their characteristics have been an area of continued research. The goal of this work is to develop a robust algorithm, which can predict hydrodynamical characteristics and dynamic stiffness and damping coefficients of hybrid and hydrostatic bearings with increased accuracy and which can be used for engineering/design purposes. The developed approach is based on Reynold's equations, where the unknown parameters are the rotor position and fluid pressure in recess pockets. Finite difference method in combination with the successive over-relaxation algorithm is used for a numerical solution of Reynold's equations. Newton's method is applied to solve the generated system of equations. Applying the developed approach, the effect of load influence on the hydrodynamical and the dynamic stiffness characteristics has been studied. Several hydrostatic bearing designs which are based on the published data were considered to compare the results calculated applying the approach with the experimental and theoretical data given in the literature. Performed study shows when journal eccentricity can't be neglected while simulating hydrostatic bearing characteristics. Simulations also allow for analysis of how different design/geometrical parameters and initial conditions (supply pressure) influence bearing performance characteristics. The developed approach can be utilized as a practical tool which allows for the prediction of performance characteristics of hydrostatic bearing with increased accuracy.

Use of high power steam turbines in maneuver regimes by power plants became a widely-distributed practice in chase of short-term economic benefits. At the same time, these actions resulted in much higher levels of lifetime consumption for turbines which are not designed for high numbers of start-ups. The purpose of the present study is to improve operational flexibility for 325 MW steam turbine through design modifications. For the accurate simulation of rotor thermo-structural state and lifetime, an improved methodology was developed. The approach allows engineers to account for the steam film condensation process and the steam flow physics for regions with the anticipated high-stress levels at front-end seal zone, and the influence of the inter-casing space steam film condensation on the flow parameters in the front-end seals chambers. Based on the simulation results for the 325 MW supercritical steam turbine HP rotor, the design changes of the front-end seal arrangement and heating conditions modification during the pre-warming phase are proposed. The results show that the proposed changes make it possible to provide a more uniform heating and lower thermo-stress level for the high-pressure cylinder rotor at the front-end seal region during the pre-warming phase, which results in an increased allowable number of turbine start-ups. The influence of heating conditions on thermo-stresses and low cycle fatigue lifetime for the baseline and modified designs as well as modeling details for transient thermo-structural analysis have been discussed.

In this study, optimal designs of hydrodynamic journal bearings for 13.5 MW induction motor prototype is developed based on the design of experiment approach and best sequences method which involves entire rotor-bearing system multidisciplinary simulations. These simulations consist of bearing hydrodynamic characteristics calculation and optimization and rotor dynamics analyses for a rotor-bearing system. The results of rotor dynamics analyses are taken into account as the constraints during optimization. Several journal bearings such as plain cylindrical with a different configuration of pockets, elliptical type, and 4-lobe fixed pad have been considered to select the most appropriate design for the application. The bearing clearance, length, diameter, pockets positions, lobe width, oil viscosity, are applied as design input variables. To find the bearing optimal design, following objective functions were considered: 1) Minimum oil film thickness. Optimal bearing clearance is designed to produce the maximum possible level of minimum oil film thickness in order to avoid or reduce possible metal-to-metal contact; 2) Maximization of the performance is done by minimization of friction power loss. 3) Rotor dynamics simulation for the rotor-bearing system is embedded in the optimization process in order to avoid resonances by providing sufficient critical speeds separation margins from operating speed. The methodology for the bearing simulation is based on the mass-conserving mathematical model, proposed by Elrod & Adams and numerical solution for the equations is generated using finite difference method. Rotor dynamics analyses are performed using finite element method. As the result of the study, optimized bearing designs for 13.5 MW induction motor were generated. Optimized bearings provide sufficient frequency margins for critical speeds for the rotor-bearing system and, at the same time, improved hydrodynamic bearing characteristics: maximized oil film thickness and increased efficiency compared to the starting design. Through the considered bearings examples, the study shows how different parameters, such as bearing clearance, length, diameter, and etc., influence key performance characteristics like bearing minimum oil film thickness, friction power losses, rotor bearing system critical speeds.

Rotor lifetime and safety primarily depend on the level of rotor vibration. In order to avoid unwanted consequences for the plant due to rotor damage and to meet the highest requirements of design reliability, accurate rotor dynamic predictions are mandatory. Having the correct rotor model is a critical issue in dynamics prediction. Often research activities are focused only on the rotor-bearing system analysis. However, generally, the whole system, which includes the rotor, bearings, casing and structural supports should be considered. Special attention should be paid to the influence of structural supports which reveals when the rotor is supported by ball bearings because of low damping and high bearing stiffness. The approach presented in this paper allows us to simulate the influence of structural supports on rotor dynamics response and as a result, the full picture of rotor-bearing-support system resonances can be analyzed to avoid potential problems. The methodology is based on support vibrations modal reduction technics. According to the approach, the natural frequencies and their mode shapes should be calculated for the separate support structure applying a three-dimensional finite element model and the relative displacements at bearing location points are measured. Supports' normalized modal characteristics (modal mass and modal stiffness), for each vibration mode should then be imported in a rotor dynamics algorithm for rotor unbalance response analysis. The approach allows for simulation of different types of support structures such as bearing pedestals, steel foundations, tabletop-type foundation, frame and pipe supports of arbitrary geometry, and so on. Validation based on the Jeffcott rotor model is presented. The current methodology has been applied to a single-stage compressor's rotor-bearing-support system which was manufactured and commissioned. The results of the simulations are discussed.

Виконано розрахункове дослідження теплового і термонапруженого стану ротора циліндра високого тиску парової турбіни потужністю 325 МВт на етапах прогріву і пуску із холодного стану.Для визначення нестаціонарного теплового стану, розроблена методика, що дозволяє визначити граничні умови теплообміну з високою точністю завдяки врахуванню процесу конденсації пари на поверхнях ротора і ступеня дискретизації теплових зон для призначення граничних умов.Врахування процесу конденсації у мiжкорпусному просторі дозволило точніше визначити параметри пара на елементах ущільнень ротора.Базуючись на результатах дослідження, запропоновано зміну конструкції і умов прогріву ротора в області переднього кінцевого ущільнення на етапі підготовки до пуску з холодного стану.Показана можливість зниження рівня термічних напружень і вплив умов прогріву на ресурс турбіни.Ключові слова: парова

In order to achieve the highest power plant efficiency, original equipment manufacturers (OEMs) continuously increase turbine working parameters (steam temperatures and pressures), improve components design and modify start-up cycles to reduce time while providing more frequent start-up events. All these actions result in much higher levels of thermo-stresses, a lifetime consumption of primary components and an increased demand for accurate thermo-structural and LCF simulations. In this study, some aspects of methodological improvement are analyzed and proposed in the frame of an integrated approach for steam turbine components thermo-structural analysis, reliability and lifetime prediction. The full scope of the engineering tasks includes aero/thermodynamic flow path and secondary flows analysis to determine thermal boundary conditions, detailed thermal/structural 2D and 3D FE models preparation, components thermal and stress-strain simulation, rotor-casing differential expansion and clearances analysis, and finally, turbine unit lifetime estimation. Special attention is paid to some of the key factors influencing the accuracy of thermal stresses prediction, specifically, the effect of 'steam condensation' on thermal BC, the level of detailing for thermal zones definition, thermal contacts and mesh quality in mechanical models. These aspects have been studied and validated against test data, obtained via a 30 MW steam turbine for combined cycle application based on actual start-up data measured from the power plant. The casing temperatures and rotor-stator differential expansion, measured during the commissioning phase of the turbine, were used for methodology validation. Finally, the evaluation of the steam turbine HPIP rotor lifetime by means of a low cycle fatigue approach is performed.

Market requirements for faster and more frequent power unit start-up events result in a much faster deterioration of equipment, and a shorter equipment lifespan. Significant heat exchange occurs between steam and turbine rotors during the start-up process and even more intensive heat exchange takes place during the condensation phase in cold start-up mode, which leads to further thermal stresses and lifetime reduction. Therefore, the accuracy of lifetime prediction is strongly affected and dependent on the accuracy of transient thermal state prediction. In this study, transient thermal and structural analyses of a 30 MW steam turbine for a combined High and Intermediate pressures (HPIP) rotor during a full cold start cycle is performed and special attention is paid to initial start-up phase with ‘condensation’ thermal BC. All steps for rotor design and the thermal model preparation were done using the AxSTREAM™* software platform. It included the development of a two dimensional model of the rotor, thermal zones and corresponding thermal boundary conditions (heat transfer coefficients and steam temperatures) calculation during turbine start-up and shut down operation. Rotor thermal and structural simulations were done using commercial FE analysis software to evaluate the thermo-stress-strain state of the turbine rotor. Calculation and validation of thermal and structural state of the rotor was done using actual start-up cycle and measured data from a power plant, and it showed good agreement of the calculated and the measured data. Based on the results of thermo-structural analysis, the evaluation of rotor lifetime by means of a low cycle fatigue approach was performed and presented in this paper.

Для расчета цилиндрических оболочек, подкрепленных продольными ребрами жесткости, предложен полуаналитический метод конечных элементов. С помощью предложенного метода численно исследуются свойства напряженно-деформированного состояния оболочек.

The effect of initial imperfections on the parametric vibrations of cylindrical shells is analyzed. The shell has moderate amplitudes of vibrations; therefore, geometrically nonlinear theory is used. The shell vibrations are described by the Donnel equations. The interaction of three pairs of conjugate modes is considered in the analysis. Therefore, the shell vibrations are described by six-degrees-of-freedoms nonlinear dynamical system. The multiple scales method and the continuation technique are used to analyze the system dynamics. The role of initial imperfections in nonlinear dynamics of shell is discussed using frequency responses.

We consider a problem of nonlinear parametric vibrations of a cylindrical shell. The vibrations are described by the Donnell–Mushtari–Vlasov equations. Motions are expanded in the modes of natural vibrations of the shell. A dynamic system with six degrees of freedom is obtained by the Bubnov–Galerkin method. The system is investigated by the method of multiple scales. The solution of the problem is represented with the use of amplitude–frequency characteristics.

The nonlinear parametric vibrations of cylindrical shell are described by the Donnell–Mushtari–Vlasov equations. The motions are represented as a mode expansion. Discretization is performed using the Bubnov–Galerkin method. The describing-function method is used to study traveling waves and nonlinear normal modes in systems with and without dissipation

Donnel's equations are used to predict nonlinear vibrations of cylindrical shells, which are excited by parametric dynamical load. A multi-degree-of-freedom dynamical system of cylindrical shells is derived. The nonlinear modes of the parametrically excited system are treated. The analyses have been carried out both with and without dissipation, using the Harmonic Balance Method. These nonlinear modes correspond to the standing waves in the shell. Traveling waves are also analyzed in detail. We come to the conclusion that the behavior of the nonlinear modes and the traveling waves are similar.

Nonlinear modes of snap-through motions of a shallow arch are analyzed. Dynamics of shallow arch is modeled by a two-degree-of-freedom system. Two nonlinear modes of this discrete system are treated. The methods of Ince algebraization and Hill determinants are used to study stability of nonlinear modes. The analytical results are compared with the data of the numerical simulations.