Rotordynamic System Research Articles

Numerical investigation of rotor-dynamic system on taking into account gas in labyrinth seals clearances

To transport natural gas over long distances, it is necessary to build gas compressor stations that compensate for gas energy losses due to friction. The natural gas uninterrupted supply to consumers depends on their work reliability. The contemporary trend towards increase performance and a simultaneous decrease in structural rigidity lead to previously unpredictable compressor rotor oscillations. The article describes the primary causes of compressor rotor vibrations and some methods of dealing with them. Limitations of existing approaches are overcome by applying multiphysicsmodeling of a fluid and structure interaction using the 2FSI approach. Modeling performed for a labyrinth-seal compressor rotor model. ANSYS software product was chosen to solve the problem, which implements the 2FSI method. The modeling was carried out on a high-performance computer complex PNRPU. Trajectories of a point on the shaft rotation axis was analyze for the solid model. The numerical modeling results of the solid structure taking into account a gas dynamics are presented. The 2FSI calculation results are compared with the solid dynamic modeling results. An imbalance influence on the system dynamic is noted. The simulations 2FSI series were performed to study geometric, kinematic and gas-dynamic parameters influence on the rotor dynamic state. The maximum effect is exerted by an initial pressure in the seal gap. The rotor and gas oscillations resonant frequency was found, which corresponds to a change in the shaft axis spatial position in the 2FSI statement.

Bifurcations of Limit Cycles in Rotating Shafts Mounted On Partial Arc and Lemon Bore Journal Bearings in Elastic Pedestals

Abstract The paper investigates the bifurcations encountered in a simple rotor dynamic system interacting with nonlinear impedance forces, generated by the supporting journal bearings of realistic profile geometry. Bearing configurations of finite arc length and of finite width, as implemented in standard design of turbomachinery have been selected, namely the cylindrical partial arc and the elliptical (lemon) bore profile. The way in which the key design parameters influence the stability of elastic or rigid Jeffcott rotor is discussed. In the scope of this study, the following bearing design parameters are considered: arc length, length to diameter ratio, geometric preload and offset, and properties of the supporting pedestal by codimension-two studies. The bearing model is coupled to a six degree of freedom shaft-disc-pedestal model with nonlinear forces calculated from the journal kinematics, bearing design and operating conditions by numerical evaluation of the Reynolds equation for laminar, isothermal flow on a two dimensional mesh. An autonomous system of differential equations is implemented. Stability of fixed points and of limit cycles for this system is evaluated applying numerical continuation. The results confirm that minor variations in journal bearing design and pedestal properties have the potential to render substantial changes in the quality of stability and the bifurcation set of the rotor dynamic system. Specific bearing profiles render significant increment of instability threshold speed while at the same time supercritical Hopf bifurcations can be shifted to subcritical with resulting instability envelopes to be generated at speeds lower than the threshold speed.

Study on the mesh stiffness and nonlinear dynamics accounting for centrifugal effect of high-speed spur gears

The centrifugal effect is one of the most common phenomena in rotor dynamic systems, while typically ignored in the mesh stiffness evaluation by many research papers. Focusing on the centrifugal force in high-speed working condition, this paper developed an analytical-FEM framework to integrate the centrifugal field into the mesh stiffness and nonlinear dynamics. The additional potential caused by centrifugal force is condensed in the mesh stiffness calculation. The gear torsional dynamic model is further extended to embody the internal dynamic effect in centrifugal field. Solutions of gear statics and dynamics are compared with existing models and experiments, and results demonstrate the reasonable accuracy of the present model.

Model and experimental verification of a four degrees-of-freedom rotor considering combined eccentricity and electromagnetic effects

In this paper, the combined eccentricity includes the mass eccentricity, static radial eccentricity and angular eccentricity in the rotor of a permanent magnet synchronous motor, and their electromagnetic effects are considered. Approximate analytical expressions of the unbalanced magnetic pull and the additional transverse electromagnetic moment are developed for the combined eccentricity. Moreover, a four degrees-of-freedom rotor system dynamics model is established that describes the influences of these electromagnetic forces and the moments on the rotor with combined eccentricity, when it is not mounted at the midspan of the rotor shaft. The comparisons of the degenerate cases of the four degrees-of-freedom model and Jeffcott rotor model with the radial and angle eccentricities are employed, and the simulation results are in good agreement with one another; further, a critical speed analysis is performed. For experimental validation, the operating mode of the rotor is selected carefully. The rotor of the permanent magnet synchronous motor in an electric vehicle is more compact in the axial direction and hence is considered to be a rigid rotor for which the critical speed is above the operating range of the angular velocity; however, half of the critical speed could be located in this range. Thus, super harmonic resonance occurs, which is discussed based on the combined eccentricity model, and the dynamic response is verified by experiments in which the results agree well in terms of the trajectory shape of the rotor compared with the results of the dynamics model.

Dynamic Responses of the Aero-Engine Rotor System to Bird Strike on Fan Blades at Different Rotational Speeds

To study the effect of a bird striking engine fan on the rotor system, a low-pressure rotor system dynamic model based on a real aero-engine structure was established. Dynamic equations were derived considering the case of the bird strike force which transferred to the rotor system. The bird strike force was obtained from the bird strike process simulation in LS-DYNA, where a smoothed particle hydrodynamics (SPH) mallard model was constructed using a computed tomography (CT) scanner, and finite element method (FEM) was used to simulate the bird strike on an actual fan model. The dynamic equations were solved using the Newmark-β method. The effect of rotational speeds on the rotor system dynamics after bird strike was investigated and discussed. Results show that the maximum bird impact force can reach 104 kN at 3772 r/min. Impact time is only 0.06 s, but the bird strike on fan blades lead to a transient shock on the rotor system. Under the action of transient shocks, the rotor system displacement in the horizontal and vertical directions increase sharply, and the closer the mass point is to the fan, the more it is affected; the vibration amplitude at the fan will increase 15 times within 0.1 s of the bird strike and will gradually decrease with the effect of damping. The dynamics of the rotor system changes from a stable single periodic motion to a complex irregular quasi-periodic motion after a bird strike, and the strike force excites the first-order vibrational mode of the rotor system. This phenomenon occurs at all speeds when bird strikes occur. Bird strikes will cause resonance in the rotor system, which may cause damage to the engine. It was also seen that the bird strike force, and hence the effects on the rotor system, increases as the engine rotational speed increases; the peak force is larger and the number of peaks has increased. The impact force at 3772 r/min is 99.5 kN higher than at 836 r/min, and three additional peaks emerged. This effect is more reflected in the amplitude, and the overall vibration characteristics do not change. Combining the bird strike with the rotor dynamics calculation, the dynamic response of the aero-engine rotor system to bird strike is studied at different flight stages, which is of guiding significance for power evaluation of aero engines after bird strike.

Unbalance identification method based on SINDy applied to an SFD rotordynamic system

In recent years, there has been an increasing interest in Data Science and Machine Learning in different topics like financial and health, this have led to start using these methods on engineer applications. This paper is focus on identify the equivalent unbalance on Squeeze Film Damper – SFD bearing using a recent machine learning technique “Sparse Identification of Nonlinear Dynamics – SINDy”. Four different cases will be examined from Bonello’s work, all of which we introduce 4 different conditions of noise to the acceleration of the system. The data for this work was obtained via a simulation of the SFD system reported on Bonello’s thesis. From the simulation only the last 20 cycles were used to feed the SINDy. This study uses a combinatorial polynomial search space over preselected functions with the purpose to identify the equivalent imbalances. Both hyperparameters: the degree of the combinatory k and the threshold value λ remaining static during all the study. There was no error between the original equations and the identified system.

Reduction of the unbalance excited vibrations during the resonance passage of a rotor

Many rotor dynamic systems are faced with large oscillation amplitudes while passing through the resonance. The coupled differential equations for the position of the center of the shaft and the rotation angle describe the dynamic behaviour of the rotordynamic system. With an appropriate external excitation the amplitude of the oscillations can be reduced. An unbalance excited oscillator is used to study the effects of a modulated angular velocity, where the angular velocity is prescribed as linear increasing with a superimposed harmonic function. The resulting excitation force is expressed as a series of Bessel functions of the first kind. In order to achieve small amplitudes near the resonance frequency of the system, special conditions for the ratio of the modulation frequency and the angular acceleration and the argument of the Bessel function of the first kind with integer order zero are derived. These requirements are first developed for a linear oscillator with an excitation force. Due to the analogy of the solution of the equation of motion these solutions are then applied directly to an unbalance excited oscillator. The results show smaller displacement amplitudes compared to the case with only linear increasing angular velocity. For the resulting motion the required torque is computed.

Influence of Fluid-Induced Forces on Cavitating Turbopumps Rotordynamics in Forced Whirl Experiment

As typically highlighted in the experimental results, in unshrouded axial inducers the fluid-induced rotordynamic forces are significantly dependent on the flow coefficient and do not present a quadratic behavior with respect to the whirl ratio, as commonly reported in radial pumps. Moreover these forces show destabilizing peaks that typically increase with decreasing cavitation number. The present paper illustrates the development of a theoretical model of the rotordynamic system capable of assessing the influence of fluid-induced rotordynamic forces on the dynamic response of turbopumps under cavitating/non-cavitating conditions. The implementation of the fluid-induced rotordynamic forces in the dynamic model has been proposed through the so-called F.I.R.F. model, which is applied to a test set-up for forced whirl experiments, in order to characterize the whirl natural frequencies and the relative mode shapes. The model has been successfully applied to a rotordynamic system including an unshrouded axial inducer and the corresponding computed critical speeds are summarized in the Campbell diagrams pointing out the difference between dry and wet analysis results.

Experimental Investigation of the Leaf Type Bearing Structure with Undersprings Under Dynamic Excitation

With foil bearings, rotors achieve high rotational speeds with less friction and wear. In addition, here the required space is small and no peripheral components like liquid tanks or pumps are needed. The drawback is a more complex prediction of the real behavior in rotordynamic systems. Impedance test rigs are suitable for investigating the structural-dynamic bearing properties and for validating the theoretical models. This article presents and discusses the measurement of dynamic behavior, i.e., stiffness and damping coefficients, of the structure of a leaf type bearing with undersprings. These measurements include variations in static load due to the relative displacement of the bearing and shaft as well as an attempt to explain the noticed anisotropic behavior of the bearing. This article also shows how much a controlled excitation improves the comparability across the frequency band. For this purpose, a test rig is presented that has been further developed in comparison to known literature approaches. The results show, that the loss factors of the examined bearing structure are up to 4 times bigger below lift-off compared to the operation at 60,000 rpm. Furthermore, the movement amplitudes and the static loads have a great influence on the stiffness and the damping.

Investigation on transient dynamics of rotor system in air turbine starter based on magnetic reduction gear

As an auxiliary mechanical device, Air Turbine Starter (ATS) uses compressed air as power source to start and drive the engine. It withstands the impact of high-pressure airflow during operation, which may cause collision between key components. For this reason, it is necessary to investigate the transient dynamics of ATS rotor system. However, different from the traditional dual rotor structure, ATS uses magnetic reduction gear (MRG) as a reduction unit, which involves multiple physical fields such as magnetic field and stress field, bringing challenges to transient dynamics analysis. In this paper, the magnetic interaction forces between various rotors are innovatively simplified into the form of springs, and added to the solution model to achieve the decoupling of multiple physical fields. On this basis, the transient displacement response of MRG-ATS has been analyzed using transient dynamics theory. The results indicate that the transient displacement of the rotor system has obvious characteristics of oscillation attenuation. The study reveals the feasibility of MRG-ATS application under transient shock.

Influence of lubricant film cavitation on the rotor dynamic system behaviour of an exhaust gas turbocharger rotor

AbstractThe vibration behaviour of fast rotating rotors is significantly influenced by the bearing properties. Lubricant film induced excitations can cause sub‐synchronous rotor oscillations known as oil‐whirl and oil‐whip phenomena. The non‐linear bearing properties depend primarily on the lubricant properties, kinematics of bearing partners and especially on the occurrence of cavitation. Outgassing processes lead to a two‐phase flow consisting of gas and oil, which can influence the bearing stiffness and damping respectively and consequently the rotor response behaviour. In this contribution, the oscillations of a semi‐floating ring supported turbocharger rotor are investigated under the influence of lubricant film cavitation. For this purpose, run‐up simulations are carried out under the assumption of mass‐conserving cavitation according to the two‐phase model and compared with measurements. In order to illustrate the influence of outgassing processes, a comparison is made with non‐mass‐conserving cavitation theory according to the assumptions of Half‐Sommerfeld.

Vibration Modelling and Control Experiments for a Thin-Walled Cylindrical Rotor with Piezo Patch Actuation and Sensing

This paper describes a dynamic model formulation and control experiments concerning the vibration behaviour of a thin-walled cylindrical rotor with internal piezoelectric patch transducers. Model development, validation and controller design procedures were undertaken for an experimental rotordynamic system comprising a tubular steel rotor (length 0.8 m, diameter 0.166 m and wall-thickness 3.06 mm) supported by two radial active magnetic bearings. Analytical solutions for mode shapes and natural frequencies for free vibration were first derived using a shell theory model, and these used to construct a speed-dependent parametric model for the rotor structure, including piezo patch actuators and sensors. The results confirm that the developed shell theory model can accurately capture the rotating frame dynamics and accounts correctly for frequency splitting from Coriolis effects. The model is also shown to be suitable for active controller design and optimization. Model-based H2 feedback control using the rotor-mounted actuators and sensors is shown to achieve vibration suppression of targeted flexural modes, both with and without rotation.

Prediction of the reaction forces of spiral-groove gas journal bearings by artificial neural network regression models

Abstract This paper presents neural network regression models for predicting the nonlinear static and linearized dynamic reaction forces of spiral grooved gas journal bearings. The partial differential equations (PDEs) are sampled, based on a full factorial and randomly spaced parameter set. Feed-forward neural network (FNN) architectures are developed for modeling the PDEs and therefore replacing the time-consuming discrete and iterative solution procedure used to this date. A significant speed-up factor of >103 in computation time is achieved, compared to solving the PDE numerically. Furthermore, the FNN allows for multi-dimensional interpolation, which makes global system optimization easily possible. This is demonstrated by a real-case rotordynamic system optimization. By using the neural network meta-models, a complete rotordynamic system optimization time reduction of factor 300 is achieved.

On post-resonance backward whirl in an overhung rotor with snubbing contact

Rotordynamic systems are central to many aerospace and heavy-duty industrial applications. The vibrational response of such systems is usually associated with forward whirl (FW) and backward whirl (BW) precessions. It is well known in the literature that the BW precession generally precedes the passage through the critical FW resonance precession. Therefore, it can be named as a pre-resonance BW frequency (Pr-BW). However, another kind of BW has been recently observed to be immediately excited after the passage through the critical FW resonance frequency in cracked rotors with anisotropic supports during run-up and coast-down operations. Consequently, this kind of BW can be named as a post-resonance backward whirl (Po-BW) precession. The Pr-BW and Po-BW phenomena are investigated here with an overhung rotor system that exhibits snubbing contact and stiffness anisotropy in the supports. Incorporating the snubbing moment couple into the equations of motion of the considered overhung rotor model yields a piecewise and strongly nonlinear system. Full-spectrum analysis is employed to capture the BW zones of rotational speeds in the whirl response. Wavelet transform spectrum analysis is also employed to determine the frequency content in the Pr-BW and the Po-BW zones. Three cases are considered in this numerical study to explore the effect of the support stiffness isotropy and anisotropy with active and inactive snubbing contact on the Po-BW excitation. For all cases, the Po-BW zones of rotational speeds are found. Moreover, the broadness and recurrence of the Po-BW zones of rotational speeds are more prominent for the cases of active snubbing contact. Even though the Pr-BW and Po-BW zones are excited at different shaft rotational speeds, they are found to possess nearly similar BW frequencies which are less than the FW resonance frequency of the considered system.

Nonlinear dynamics of rotor system supported by bearing with waviness.

A new nonlinear rotor model supported by the rolling bearing is established under the consideration of the bearing with waviness fault, the unbalanced excitation, the nonlinear Hertz contact force, the varying compliance vibration, and, especially, the physical nonlinear stiffness of the shaft material. The expression with cubic nonlinear terms is adopted to characterize the physical nonlinear stiffness of the shaft material, and the sinusoidal wave is applied to describe the shape characteristics of the waviness fault. The dynamic equations of motion for the new model are developed, and the calculation example of the rotor system supported by the bearing JIS6306 is solved by the variable step-size Runge-Kutta methods to study the effect of the waviness, the clearance, the mass eccentricity on the dynamic behavior. The research results show that growth of the amplitude for the waviness changes the energy distribution of the vibration process; the enlargement of bearing clearance will reduce the stability of the system; the increase in the number of the waviness will make the order of the frequency components changed; for the nonlinear stiffness bearing-rotor system with waviness fault, the augment of mass eccentricity will enhance the impact of the nonlinear stiffness on the system.

Nonlinear Responses of An Unbalanced Overhung Rotor-Short Journal Bearing System with Some Bifurcation Results

Both autonomous and non-autonomous nonlinear systems display complex responses including the transient and long run responses. An overhung rotor-short journal bearing system (ORSJB) is the excellent paradigm exhibiting nonlinear responses and of practical importance. The unbalanced overhung rotor-journal bearing system is formulated by using modified Laval-Jeffcott rotor model. The nonlinear fluid forces model of the short journal bearing is adopted and rearranged in a suitable form for numerical analysis. This paper investigates numerically a bifurcation point, transient and long run responses of the ORSJB system. The rotational speed is preferable bifurcation parameter herein. The ORSJB system without unbalanced eccentricity for the selected parameters yields the bifurcation value of Ω = 3.0091. Limits cycles in the long-run responses with eU =0.002 m and without unbalanced eccentricity at long run are presented in the bifurcation regime at the speed of Ω = 6.6869 and Ω = 0.01671 respectively. The transient response with eU =0.002 m at the speed of Ω = 4.7142is elucidated. The evident results in this paper give only a partial view of bifurcation behaviours. Bifurcation analysis is a useful tool for design and operation overhung rotor dynamic systems.

Squeeze film dampers supporting high-speed rotors: Rotordynamics

This work develops a finite element based multi-mass flexible rotor model for theoretical investigation of the influence of the squeeze film damper lubricant inertia on the unbalance-induced steady-state and transient vibration amplitudes of high speed turbomachinery. The rotordynamic model is developed by applying the principles of finite element analysis to discretize the rotor components, including the rotor shaft and disk, into local elements with mass, stiffness, and gyroscopic matrices. Subsequently, the local matrices are assembled together to develop the global model of the rotordynamic system. The influence of squeeze film damper lubricant inertia is incorporated into the model by using short-length cavitated damper models with retaining springs executing circular-centered orbits. Additionally, the rotordynamic model incorporating the nonlinear squeeze film damper models is iteratively solved in the time domain by applying a predictor-corrector transient modal integration numerical method and the steady-state and transient motions of the rotor system are investigated under different rotor and squeeze film damper parameters. The results of the study verify the substantial influence of squeeze film damper lubricant inertia on attenuating the vibrations of high-speed turbomachinery. Furthermore, the developed rotordynamic model delivers an efficient and powerful platform for the analysis of high-speed turbomachinery, including jet engines and gas turbines.

An accurate instantaneous angular speed estimation method based on a dual detector setup

Instantaneous angular speed (IAS), or ‘true IAS’, is an inherent signature directly related to the mechanical dynamics of rotor systems and has been employed extensively in different rotating machines. Due to various uncertainties, such as manufacturing error or the inaccurate installation of encoders, the acquired IAS signals often contains significant contents of ‘false IAS’ which can mask the small variations of IAS, lead to low accuracy and consequently affect the understanding of the dynamic behaviour of a rotor system. This study proposes a dual detector method to minimize the influence of encoder errors and hence obtain more accurate IAS estimation. Through theoretical study, it is proved that the false IAS induced by encoder errors from the two detectors have the same amplitudes but with a phase difference, whereas the true IAS components from the two detectors are identical. On this basis, an algorithm is developed firstly to acquire the encoder errors through a phase alignment operation and then estimate the true IAS after removing the estimated encoder errors. The proposed method is evaluated by both numerical simulations and experimental studies. The results show that the proposed method can more significantly suppress the encoder errors associated with encoder eccentricities in comparison with conventional reference methods, thereby yielding an accurate true IAS from the measurement. The proposed method can be particularly effective when the encoder has very low accuracy, allowing a more cost-effective IAS monitoring system to be achieved.

On the modeling of rotors with rolling element bearings using bond graphs

Abstract This paper describes the development of a generic rotordynamic model incorporating rolling element bearings. Aimed at examining model fidelity requirements for bearings, the work contributes to advancing knowledge on rotordynamic system models suitable for studying mechatronics- and controls-related applications, transient issues and fault conditions. The bond graph method is applied in a consistent manner, giving a clear model structure. Body-fixed equations of motion, derived from Lagrange's method, are contained in field elements. An alternative cage implementation and a flexible outer ring are introduced, providing a more complete basis for bearing modeling. The flexible outer ring is modeled using the modal approach with a solution for moving loads. The derivation of the equations of motion and the bond graph formulation are thoroughly presented. Following the display of versatility and connectivity in the modeling approach, a simulation example is presented to demonstrate the capability and usefulness of the model for transient analyses. A classic time-series analysis of a run-up of an unbalanced rotor, with and without a damaged bearing, directly yields interesting results; the complex interaction between model elements following the passing of the first critical speed leads to a stationary vibratory condition taking significantly long time to develop in the damaged bearing case and involves persistent slip.

Design optimization of a rotordynamic beam system with elastic supports to minimize flexural responses using a combined optimization algorithm

Modern rotating machines are often required to operate with increasingly limited physical footprints and/or overall weights, necessitating that they utilize wide-ranging shaft speed to accomplish needed power levels. It is not uncommon then that the resulting speed envelope crosses or encompasses multiple critical speeds with the requirement that flexural vibrations be limited to safe and acceptable levels. Design optimization of this class of problem can be particularly difficult in that responses can exhibit high modality adding significant complexity and time to the optimization effort. This paper presents a design optimization methodology appropriate to an economic solution for this class of problem. A simplified rotordynamic example is used to challenge the proposed optimization method against other more “conventional” techniques with results of the proposed method extensible to more complicated, practical rotordynamic problems.