Rotordynamic Behavior Research Articles

Probabilistic Analysis of Orbital Characteristics of Rotary Systems with Centrally and Off-Center Mounted Unbalanced Disks

Rotor dynamics plays a crucial role in the performance and safety of rotating machinery, with disk position and unbalance significantly impacting system behavior. This study investigates the dynamic characteristics of two rotor configurations: a centrally mounted unbalanced disk (Rotor05un) and an off-center unbalanced disk (Rotor025un). Using numerical simulations and Monte Carlo analysis, we examined critical speeds and orbital patterns for both configurations. Probability distributions of shaft orbital positions revealed distinct patterns for each configuration. Quantile analysis revealed approximate linear trends for Rotor025un, suggesting higher system stiffness and more predictable behavior near critical speeds. Cross-sectional analyses of the orbits provided insights into the complex interactions between disk position, gyroscopic effects, and system natural frequencies. These findings provide valuable insights for rotor system design, particularly for applications with non-ideal mass distributions. The study goes beyond traditional critical speed analysis to examine orbital patterns and point on orbit occurrence from a probabilistic perspective. Based on the simulation of the orbits, an orbital is determined that allows the probability of the shaft occurring at the analyzed distance from the origin to be determined. The paper also offers insights into the complex interaction behavior of chosen rotor configurations and highlights the importance of considering disk position in predicting and optimizing rotor dynamic behavior, contributing to the development of more robust and efficient rotating machinery.

Rotor-dynamic study for high pressure multistage centrifugal pump

Abstract Rotor-Dynamic behavior were closely related to the reliability, operability and cost effective for high pressure multistage centrifugal pump. The middle sealing sleeve, high-pressure sealing sleeve, front and rear rings of the impeller of the pump set were studied in this paper, by establishing a theoretical calculation model, numerical calculation methods. The leakage, radial force and tangential force of those seals under different working media and different sealing gaps were ccalculated in detail. The results showed that the sealing performance and dynamic characteristics were influenced by working medium and the sealing gap. The working medium had little effect on the radial force and tangential force of the middle sealing sleeve and high-pressure sealing sleeve, while with a greater effect on the front and rear rings of the impeller. The radial force of the middle shaft sleeve was closed with the sealing clearance. As the seal gap was 0.5mm, negative stiffness at low and high frequency whirlpools was more easily occured. With the gap increasing, the unbalanced response of the rotor increased significantly.

Supercritical Operation of Bearingless Cross-Flow Fan

This paper presents a decoupled bearingless cross-flow fan (CFF) that operates at a supercritical speed, thereby increasing the maximum achievable rotational speed and fluid dynamic power. In magnetically levitated CFF rotors, the rotational speed and fan performance are limited by the bending resonance frequency. This is primarily defined by the low mechanical bending stiffness of the CFF blades, which are optimised for fluid dynamic performance, and the heavy rotor magnets on both rotor sides, which add significant mass but a minimal contribution to the overall rotor stiffness. This results in detrimental deformations of the CFF blades in the vicinity of the rotor bending resonance frequency; hence, the CFF is speed-limited to subcritical rotational speeds. The novel CFF rotor presented in this study features additional mechanical decoupling elements with low bending stiffness between the fan blades and the rotor magnets. Thus, the unbalance forces primarily deform the soft decoupling elements, which enables them to pass resonances without CFF blade damage and allows rotor operation in the supercritical speed region due to the self-centring effect of the rotor. The effects of the novel rotor design on the rotor dynamic behaviour are investigated by means of a mass-spring-damper model. The influence of different decoupling elements on the magnetic bearing is experimentally tested and evaluated, from which an optimised decoupled CFF rotor is derived. The final prototype enables a stable operation at 7000 rpm in the supercritical speed region. This corresponds to a rotational speed increase of 40%, resulting in a 28% higher, validated fluid flow and a 100% higher static pressure compared to the previously presented bearingless CFF without decoupling elements.

Investigation on Pivot Rolling Motion Effect on the Behavior of Rocker Back Tilting Pad Journal Bearings

The rolling motion of the pads with rocker back (RB) and ball and socket pivots is normally neglected in common software programs for the study of the rotor dynamic behavior of tilting pad journal bearings (TPJB). In other words, the theoretical contact point of the pivot is considered fixed. The aim of this work is to provide a novel way to implement in commercial software the effect of the variation of the circumferential coordinate of the theoretical contact point due to the pad rolling motion in RB TPJB. This is done by introducing an equivalent pivot rotational stiffness evaluated with an analytically derived formula, validated through finite element analysis. Such a stiffness is a function of the pad load and the radii of the contact pair, increasing with the load, the radii, and the degree of conformity of the contact. The static and dynamic characteristics of a five pad RB TPJB are then evaluated with a commercial software with and without the rotational stiffness contribution for two different pivot geometries. Non-negligible differences were found, particularly regarding the cross-coupled dynamic coefficients that show the higher sensitivity to the rotational stiffness. The inclusion of a pivot rotational stiffness among the data of commercial software for simulation of RB TPJB could contribute to fill the gap between numerical and experimental results.

Influence of a transient bubble dynamics cavitation model for squeeze film dampers on the run‐up behaviour of a turbocharger rotor

AbstractThe force response of squeeze film dampers and hydrodynamic bearings is significantly influenced by the occurence of cavitation inside the lubricant films. The viscosity of the occuring mixture of oil and air bubbles is vastly different than that of pure lubricant and therefore changes the pressure build‐up and thereby the stiffness and damping properties of the lubrication films. To accurately predict the force response of a squeeze film damper, the cavitation and its effect on the pressure build‐up therefore has to be taken into account. Well known cavitation algorithms for lubrication problems, like the Elrod algorithm or the two‐phase‐model, neglect the transient nature of outgassing and assume instantaneous bubble formation and collapse. In the presented paper, a transient bubble dynamics model based on the Rayleigh‐Plesset‐Scriven equation is utilised instead and its effect on the rotordynamic behaviour of a turbocharger rotor is examined based on fully transient multi body dynamics simulations.

Benchmark solutions for the dynamic analysis of general rotor-bearing system with arbitrary linear boundary conditions

The rotor unbalance is a major source of rotor vibrations. Rotor vibrations often produce many undesired effects like noise, wear and fatigue, etc. In this paper, we try to seek the benchmark solutions for the unbalance responses of complex rotor-bearing system. The presented approach could be seen as the extension of transfer matrix method (TMM) in some sense. For the TMM, a disk or supporting structure cut off one uniform shaft element into two and someone must use the compatibility condition between these two new elements to derive the transitive matrix. However, for the presented approach, it directly solves the governing equations of uniform shaft elements with consideration of the effects of disks and supporting structures. Thus, this analytical approach is advantageous in reducing the times of matrix multiplication between state matrices and field matrices. One only needs to calculate the inversion of 16 × 16 dynamic stiffness matrix to find the steady state response. It saves the computer memory and is easy to be programmed. In addition, this analytical approach avoids the problems of selecting optimal discretization mesh densities in the case of FEM applications. For arbitrary linear boundary condition, the benchmark solutions are always easy to be obtained. The numerical simulation is carried out and two numerical examples are given to validate the new solutions. In which the finite element method is used as the numerical approach. Simulation results show that the benchmark solutions match very well with the FEM results. The effects of anisotropy in the supporting structures on the rotor's dynamic behavior, which are observed in this work, are also in accordance with many references. This validates the benchmark solutions further.

Design and optimization of squirrel cage geometries in aircraft engines toward robust whole engine dynamics

In this work, an coupled end-to-end approach for the optimization of the rotor dynamic behavior of a dual-spool aircraft engine along with fatigue life optimization of squirrel cages (SQC) is presented. A realistic model to simulate the rotor dynamics is created, where the high-pressure (HP) rotor is supported by two squirrel cages. The aim of this work is to find a robust rotor dynamics design by shifting a critical speed to higher rotational speed, and at the same time improving the squirrel cage design with respect to fatigue life. Fully automatized coupled analysis process chain is implemented, allowing to compute the influence of the SQCs geometry variation onto the full rotor dynamics and structural performance of the SQC. Two global optimization techniques are employed to explore the SQCs design space and find optimal 3D geometries, using the aforementioned coupled process. Optimization results are compared and discussed in detail, indicating the importance of the numerical optimization to improve fatigue life of the squirrel cage. It is shown that optimized and non-optimized SQC designs, both fulfilling rotor dynamics goals, can have significantly different performance regarding their fatigue life. Moreover, the advantage of the coupled process is illustrated, allowing to find superior SQC designs by considering both disciplines simultaneously in comparison with a sequential (uncoupled) approach, when the target elastic properties of an SQC, selected only based on the rotor dynamics requirements, may lead to sub-optimal fatigue life.

Modeling and mu-synthesis control of a flexible rotor stabilized by active magnetic bearings including current free control

Designing a controller for a magnetically stabilized rotor that experiences non negligible gyroscopic moments requires careful attention on the rotordynamic behavior, the design of the position and current free control as well as the actuator and sensor characteristics. Most of the controller design procedures for magnetic bearings separate the synthesis of the current free controller and the position controller using different methods. This article connects the synthesis of the current free controller and the position controller in the μ-framework. Therefore, the modeling of the rotor, the position controller design and the design of the current free control of a flexible rotor stabilized by active magnetic bearings is described. The rotor is modeled by using the finite element method. Afterwards, the flexible system is reduced to get an appropriate model size for the control design. For the optimization of the controller parameters the μ-synthesis method is applied. However, compared to the general μ-synthesis method in this article an a priori fixed control structure is utilized. This control structure splits up the tilting subsystem to a control path for the nutation mode and one for the precession mode. Thus, the proposed controller has the advantage of a low system order and that the parameter variant nature of the plant is already considered in the control structure. Additionally, the synthesis of the current free controller is added to the μ-framework. Hence, the current free controller is optimized for a plant with defined uncertainties using necessary constraints of the closed loop systems. Finally, the stabilization of the system using the proposed control method is experimentally validated on a turbomolecular pump with a magnetically levitated rotor.

Manufacture and Testing of a Magnetically Suspended 0.5-kWh Flywheel Energy Storage System

This article presents crucial issues regarding the design, manufacture, and testing of a steel rotor for a 0.5-kWh flywheel energy storage system. A prototype was built using standard industrial components. The rotor has a maximum operating speed of 24 000 min <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1</sup> and is magnetically suspended. The introduced critical issues regarding the manufacture include the thermal stress during the rotor hardening process, the fixation for the rotor components of the drive machine and the magnetic bearings, etc. The rotor strength was validated by the overspeed testing according to the standard <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IEC 60034-1</i> . A rotation of up to 28 800 min <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1</sup> (120% of the maximum speed) was realized and maintained for 2 min. The proved rotor was suspended by commercial magnetic bearings in the flywheel system. Despite simple decentralized control, the rotor was successfully driven up to the designed maximum speed. The rotor dynamic behavior is analyzed by analytical calculation, simulation, and measurements. Different eigenmodes are identified. The results prove the possibilities to build and operate a flywheel system using standard industrial components, showing potential for cost-effective designs.

Dynamic Characteristics of a 5-kN Thrust Gas Turbine Front Bearing Composed of a Vibration Absorber Element and Deep Groove Ball Bearing

Rolling bearings enable the development of high rotational speeds with a good load-bearing capacity and low frictional resistance. Therefore, rolling bearings can be used on a wide variety of rotating machines. This study investigated the dynamic stiffness and damping coefficients of the front bearing of a 5-kN thrust gas turbine. A Barden® 206(T) deep-groove ball bearing was the main part of the front bearing. Computational models were developed using MATLAB® for both the structural dynamics problem governed by Hertzian contact theory and the lubricated problem governed by elastohydrodynamic theory. In addition, the front bearing was composed of a vibration absorber element, which was also considered. Finite element analysis was performed on both the ball bearing and the vibration-absorbing element to assist in its final characterization. As the main results, the stiffness and damping dynamic coefficients of the front bearing were estimated and can be used to predict the rotordynamic behavior of the aeronautical gas turbine rotating set to avoid possible vibration problems during operation.

Influence of Fluid Film Bearings with Different Axial Groove Shapes on Automotive Turbochargers: An Experimental Study

Most commercial automotive turbochargers (TC) employ semi-floating ring bearings (SFRB) with axial groove shapes. In order to bring some insights into the role played by the axial groove geometry on the dynamics of TC, this work deals with an experimental study of the rotordynamic behavior of a stock automotive turbocharger operating on SFRB with two different groove shapes, which have the same volume and width, and with the same number of grooves. The rotating machine behavior has been evaluated under different operating conditions using a test bench specially designed to analyze turbochargers. Rotordynamic (RD) characteristics of automotive turbochargers are estimated to evaluate the influence of the axial groove geometry on the machine vibratory behavior. Frequency spectra and orbital plots of the rotor are obtained from accelerometers and proximity probes mounted on the turbocharger. The comparative analysis of the vibrational behavior of automotive turbochargers running on different supporting systems allows the identification of the role played by the axial grooves on the machine rotordynamic performance. The experimental results rendered in this work permit to classify the influence of the axial groove geometry on the turbocharger rotordynamic behavior for several speed and flow conditions.

Continuous Rotor Dynamics of Multi-Disc and Multi-Span Rotor: A Theoretical and Numerical Investigation on the Identification of Bearing Coefficients from Unbalance Responses

Identification of bearings’ stiffness and damping coefficients, which strongly affects the dynamic characteristics of rotors, is another inverse problem of Rotor Dynamics. In this paper, aiming at multi-disc and multi-span rotors, two novel algorithms are proposed for identifying each bearing’s coefficients based on the continuous rotor dynamic analysis method. A linear functional relationship between the main complex coefficients and the cross-coupled complex coefficients is obtained, which eliminates the coupling between the coefficients and the rotor unbalance in the forward problem. Then, Algorithm I is proposed. However, it is only suitable for rolling-bearing. To solve the problem, changing the rotating speed slightly is proposed to solve the difficulty that another set of equations cannot be developed because the slope of the proposed linear function is constant when the rotating speed is maintained at a fixed speed. Then, Algorithm II, which can be applied to both rolling-bearing and oil-journal bearing, is provided. Numerical investigations are conducted to study the two methods. It is indicated that there should be a measuring point, called an adjustment point, near each bearing, whose coefficients should be identified, to obtain high identification accuracy. Moreover, the identification accuracy of the two algorithms is strongly related to sensor resolution. When the measuring errors of all the required unbalance responses are zero or the same, the identification errors are almost equal to zero. In conclusion, the proposed algorithms provide a method for monitoring the stiffness and damping coefficients of all bearings in a multi-disc and multi-span rotor under operation conditions to predict rotor dynamic behavior for the safe and steady running of rotating machines.

Numerical modeling on friction and wear behaviors of all-metal progressive cavity pump

The wear of the all-metal progressive cavity pump (AMPCP) has become one of the main concerns from various field applications of oil production. This work systematically reveals the mechanisms of the wear and lubrication between the stator and the rotor components of the AMPCP when lifting fluids. An efficient simulation strategy is proposed to couple the two simultaneous processes, including the rotor's dynamic behavior and material loss due to wearing. On this basis, a three-dimensional finite element model based on the explicit dynamic method is established to capture the rotation process of the rotor considering the pressure load of the inner fluid. In addition, the frictional contact and wear behavior between stator and rotor is described as well. In this regard, the wear estimation model based on the Archard wear theory is developed to depict the material loss with a geometry update and local remeshing technique, the reliability of which is well verified against experimental results. Based on the statistical results of characteristic contact time (CCT) and characteristic wear band (CWB), the wear pattern is summarized into three distinctive bands denoted by CWB I, CWB II, and CWB III, which occurs at different stages during the operating cycles of AMPCP. In particular, the influence of operational parameters (e.g., rotation speed, clearance, and lubrication) on the wear behavior and its underlying mechanisms are quantitatively revealed and discussed. The results demonstrate that the lubrication effect could substantially reduce the wear degree to extend the service life of AMPCP.

Determining the Influence of Casing Vibrational Behavior on Rotordynamics

Abstract Casings of machinery and support structures have an influence on the rotordynamic behavior, which is commonly considered by simplified models (e.g., one degree-of-freedom models). These are in many cases insufficient. Hence, more accurate modeling approaches are required, which can be used in the design process or the rotordynamic calculation to achieve a better representation of the overall vibrational behavior. To study the effects of casing and supporting structures on rotordynamics, the casing modal parameters of an axial compressor are determined by an experimental modal analysis. In parallel, a numerical model is established. As experimental data are rarely found in the literature, this work focuses on the parameter identification of the casing structure. The results are subsequently incorporated into a model updating strategy, in order to tune and improve the numerical model. Experimental and numerical data are compared to assess the quality of the data and the results gained. The ultimate objective is a reduced order model, which can be integrated in existing rotordynamic tools via an interface while keeping the calculation time low.

Tilting Pad Journal Bearing Ball and Socket Pivots: Experimental Determination of Stiffness

Tilting pad journal bearings (TPJB) are used in turbomachinery for their stability at high speeds. For design purposes, it is necessary to preliminarily investigate the turbomachine rotor dynamic behavior by simulation. The dynamic characteristics of all components must be known as precisely as possible and experimental validation of each single model is required. While a lot of work has been carried out on bearings, the ball-and-socket stiffness is still estimated by means of Hertzian formulas. Recently, some authors have used the finite element method, but it seems that nothing has been done experimentally to date. This paper describes the test rig designed to determine the stiffness of a TPJB ball-and-socket pivot by equipping the grippers of a tensile universal testing machine with specifically designed interfaces. A methodology for evaluating the stiffness from the experimental results is reported. Preliminary compression results are presented and compared with the analytical ones obtained using Hertz’s formula showing significant differences for the ball-and-socket conformal contact.

Approximation of Non-Linear Rotor Dynamic Resonance Behavior of Vertically Aligned Hydro-Units Guided by Tilting-Pad Bearings

Dynamic analyses of vertical hydro power plant rotors require the consideration of the non-linear bearing characteristics. This study investigates the vibrational behavior of a typical vertical machine using a time integration method that considers non-linear bearing forces. Thereby, the influence of support stiffness and unbalance magnitude is examined. The results show a rising influence of unbalance on resonance speed with increasing support stiffness. Furthermore, simulations reveal that the shaft orbit in the bearing is nearly circular for typical design constellations. This property is applied to derive a novel approximation procedure enabling the examination of non-linear resonance behavior, using linear rotor dynamic theory. The procedure considers the dynamic film pressure for determining the pad position. In addition, it is time-efficient compared to a time integration method, especially at high amplitudes when damping becomes small.

Deformations of a piezo‐actuated bearing ring in journal bearings

AbstractAn elastically deformable bearing ring is investigated. The piezo‐actuated shaping of the elastic bearing ring as well as the journal motion are coupled and determine the height of the lubrication gap which takes a crucial part in the theory of hydrodynamic lubrication. The influence of the active‐dynamic shaping of the bearing ring on the rotor dynamic behaviour has been investigated in [1], especially for a two‐lobe journal bearing. The present investigations are aiming at the active‐static shaping of a bearing ring, i.e. a finite number of piezo actuators is used to cause deformations of the bearing ring. Moreover, the realisation of different shapes by the introduced piezo actuators is examined, whereas the desired shapes are not necessarily limited to two‐lobe geometries.

Numerical calculation of the rotordynamic coefficients of a LOX-Turbopump Seal for the LUMEN LOX/LNG demonstrator rocket engine

In rocket engines seals are one of the key elements for a successful turbopump operation. They are not only responsible for reducing the leakage but they also have a huge influence on the rotordynamic behaviour. Depending on the design, the seal on a turbopump impeller can damp or excite rotordynamic instabilities. In this work we present transient 3D CFD simulations of straight gap seals, which are used to determine the rotordynamic coefficients. After that the numerical simulation is extended to characterize the influence of the impeller to housing clearance and swirl brakes on the rotor dynamic coefficients. We show that the inertia term cannot be neglected for the investigated nominal rotational speed of the rotor and high absolute whirl frequency ratios especially for the rotor housing clearance. Finally the coefficients calculated by numerical simulation are used in the equations of motion for a floating ring seal and the equations are solved numerically to do a sensitivity study of a floating ring to variations in axial force, rotor displacement and floating ring mass.

Effects of bearing lubrication conditions on rotor dynamic behavior

Usually, rotor dynamic analyses do not consider the effects of the lubricant supply condition, although this is an important aspect of journal bearing performance. Therefore, this work has implemented a mass-conserving model to realistically describe the oil feeding condition and evaluate its influence on rotor dynamics. This analysis was carried out by varying feeding pressure and groove position and the results indicate that the oil supply conditions significantly influence the bearing performance and affect the dynamic behavior of the entire rotor. In general, lower flow rates lead to a bearing with a greater stability region, as observed mainly in the bearing with a groove at 270° and, on the other hand, the increase in feeding pressure reduces the instability threshold.

Rotordynamic Behavior of Leaf Seals: Measurement of Frequency-Dependent Force Coefficients for Varying Inlet Parameters

Abstract While brush seals can be found in various applications for turbomachines today, leaf seals are a further development in compliant seal technology and have a lower level of maturity. Among the purported advantages are greater axial rigidity when subject to higher pressure differences and the potential for noncontacting operation due to lift-up. However, especially their rotordynamic behavior is little investigated in the literature so far. In this paper, measured rotordynamic force coefficients of a leaf seal are presented for varying inlet pressures, preswirl velocities, and excitation frequencies. The leaf pack of the tested leaf seal has zero rotor cold clearance, and its coverplates are designed for facilitating a lift-up effect when pressurizing the seal. Experiments were performed on a dynamic test rig with whirling rotor using active magnetic bearing (AMB) technology and evaluated in the frequency domain based on the impedance method. Test results for the leaf seal reveal positive direct stiffness and an advantageous rotordynamic behavior due to significant levels of direct damping and negative cross-coupled stiffness throughout the operating parameter range. Leaf seal results are compared to brush and labyrinth seal data from previous studies for varying inlet pressures and preswirl velocities. Additional computational fluid dynamics (CFD) simulations were carried out to predict the leaf deflection moment, which supports the findings regarding hydrostatic lift-up from the experimental results.