Threshold Speed Of Instability Research Articles

Dynamic characteristics of a flexible rotor supported on surface textured journal bearings

Abstract This paper presents the dynamic characteristics of a flexible rotor supported on surface textured journal bearings. Oil lubricated journal bearings exhibit the self-exited whirling of rotor, leading to instability. Understanding the threshold speed of instability is crucial in mitigating excessive vibration in rotating machinery. In this work, a simple model of a flexible rotor with central disc is studied. Modal analysis is carried out to obtain the whirl frequencies of the system, mode shapes, and stability limit speed. The addition of surface textures to the journal bearing stands out as a significant modification aimed at improving performance. Fully textured and partially textured bearing surfaces are considered in the investigation and compared with the performance of smooth journal bearings. Notably, fully textured bearings enhance stability limits. This study offers insights essential for optimizing machinery performance and minimizing operational risks.

Vibration and bifurcation analysis of rotor systems with air ring bearings including ring tilting

Air ring bearings may be considered as a further development of classical air bearings. A drawback of classical rigid air bearings is their reduced damping behavior, which usually prevents a secure operation above the linear threshold speed of instability. Considering air ring bearings, the ring-shaped bearing bushing is visco-elastically mounted in the housing so that external damping is provided. Alternatively, the bushing ring may also be mounted in the housing by a foil structure (beam or bump foil design) so that dissipation is generated by dry friction. Due to the external damping/friction provided by the ring mounting, rotor systems with air ring bearings can be operated above the threshold speed of instability so that stable self-excited vibrations with moderate amplitudes are observed. For certain system parameters, however, the rotor system can bifurcate into a rather dangerous vibration mode, where bearing ring modes are excited by the whirl/whip frequency of the air films. To accurately predict the bifurcation into the problematic ring modes, ring tilting has to be taken into account.Here, a detailed analysis for rotor systems with air ring bearings is presented, which fully incorporates ring tilting effects. The influence of ring tilting on the stability and bifurcation behavior is discussed in detail with the help of nonlinear rotor/bearing models. In connection with the bifurcation into the dangerous ring mode, an interesting nonlinear mode coupling effect (1:2 mode synchronization) is detected. A detailed analysis and a clear physical explanation of this mode coupling phenomenon is given.

Numerical investigations on high-speed turbo-compressor rotor systems with air ring bearings: Nonlinear vibration behavior and optimization

Regarding aerodynamic bearings, mainly two basic types can be distinguished: air bearings with a rigid housing and air foil bearings with an elastic supporting structure. Here, a modified bearing concept -- called air ring bearing -- is investigated, which may be considered as a further development of rigid air bearings or as a combination of the two basic bearing types. The idea of air ring bearings is to insert a ring-shaped bearing bushing between the shaft and the foil structure. Alternatively, a visco-elastic supporting structure (e.g., an elastomer) can be applied to connect the bushing ring with the housing. In the first case, external dissipation is mainly generated by dry friction. In the latter case, viscous damping is used to provide external dissipation. Due to the external friction/damping generated by the ring mounting structure, rotor systems with air ring bearings can be operated above the linear threshold speed of instability so that stable self-excited vibrations with moderate amplitudes can be achieved in the complete speed range. Here, a detailed transient co-simulation model for rotor systems with air ring bearings is presented. The rotor is represented by a multibody model. The air films of the two bearings are represented by two nonlinear time-dependent finite element systems. The multibody model and the two finite element subsystems are solved simultaneously by means of an explicit sequential co-simulation technique. Due to the strong nonlinearities, the system shows interesting vibration and bifurcation effects, which are investigated in detail with the help of run-up simulations.

Analytical and experimental investigation on stability of rotor system with spline coupling considering torque, friction coefficient and external damping

The spline coupling is widely used in high-speed rotor systems. As the lack of good understanding of the instability and influence behaviors, the spline induced instability is still not well controlled. From the view of dynamics, torque, friction coefficient and external damping are key factors that influence the stability directly. In this paper, the mechanism and influence law of spline induced instability are investigated. A new model to predict the threshold speed of instability of spline coupled rotor accurately is proposed. A criterion to determine the stability of rotor through judging the relative displacement between splines is derived. Then the effects of torque, friction coefficient and external damping on stability are investigated. Corresponding experiments are carried out to validate the model. The characteristics of instability caused by spline are observed successfully through repeated experiments. Results demonstrate that the mechanism of instability caused by spline is the relative effect of the internal damping from spline coupling and the external damping. This investigation can help with the prediction of instability caused by the spline and the regulation of rotor stability.

Nonlinear Dynamics of Turbine Generator Shaft Trains: Evaluation of Bifurcation Sets Applying Numerical Continuation

Abstract The nonlinear dynamics of turbine generator shaft trains for power generation are investigated in this paper. Realistic models of rotors, pedestals, and nonlinear bearings of partial arc and lemon bore configuration are implemented to compose a nonlinear set of differential equations for autonomous (balanced) and nonautonomous (unbalanced—per ISO) cases. The solution branches of the dynamic system are evaluated with the pseudo-arc length continuation programed by the authors, and the respective limit cycles are evaluated by an orthogonal collocation method and investigated on their stability properties and quality of motion for the respective key design parameters for the rotor dynamic design of such systems, namely, bearing profile and respective pad length, preload and offset, pedestal stiffness and elevation (misalignment), and rotor slenderness. Model order reduction is applied to the finite element rotor model and the reduced system is validated in terms of unbalanced response and stability characteristics. The main conclusion of the current investigation is that the system has the potential to develop instabilities at rotating speeds lower than the threshold speed of instability (evaluated by the linear approach) for specific unbalance magnitude and design properties. Unbalance response (with stable and unstable branches) is evaluated in severely reduced time compared to this applying time integration methods, enabling nonlinear rotor dynamic design of such systems as a standard procedure, and revealing the complete potential of motions (not only local).

Stability Analysis of Rotor with a Spline Coupling

Abstract The spline coupling is widely used in aero-engines. Once the spline coupling fails, it will cause great damage to the engine. In this paper, the stability of a rotor with a spline coupling was studied. The internal damping instability model of the rotor with spline coupling is deduced, and the instability mechanism is explored. The stability boundary is analyzed by using the R-H criterion. The mechanism of rotor instability is analyzed by simulation. An experimental system of rotors with different structural parameters of spline coupling was built. The influence of the positioning surface clearance on the stability was analyzed. The modeling results show that the internal friction of the spline coupling will introduce additional damping and anti-symmetrical cross stiffness to the rotor, the additional damping will reduce the vibration response, and the anti-symmetrical cross stiffness will cause the rotor to become unstable. The simulation and experimental results show that the rotor system will not be unstable when the two positioning surfaces of the spline coupling are in an interference fit. When one of the positioning surfaces is a clearance fit and the other is an interference fit, the internal friction will cause the rotor to become unstable. The instability threshold speed is higher than the first-order critical speed. At the same time, due to the additional damping introduced by the sleeve tooth structure, there will be a transition stage of instability. At this time, the rotor will vibrate with sub-harmonic components, but the vibration amplitude of the rotor will decrease. When the two positioning surfaces are clearance fit, the rotor is unstable and the amplitude increases suddenly. The obtained instability characteristics of the rotor with spline coupling have important value for the instability fault diagnosis, and provide help for the stability control.

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.

Modal characteristics and dynamic stability of a whirling rotor with flexible blades

This paper analyses the modal characteristics and dynamic stability of a whirling rotor with flexible blades. A rigid disk with flexible blades, which is called a rotor-blade system in this paper, is modelled as a Jeffcott rotor with Euler-Bernoulli beams. From the nonlinear equations of motion, we derived the equilibrium position and the linearized equations around the equilibrium position by the perturbation method. Based on the linearized equations, a modal analysis was performed for the variations of the supporting stiffness and rotating speed. Furthermore, we investigated the effects of the whirling motion on the natural frequencies and mode shapes and found the threshold speed of instability that cause instability in the system. In addition, we also studied the characteristics of stable and unstable regions in terms of the rotating speed and the supporting stiffness.

Investigation on the rotordynamic performance of hybrid bump-metal mesh foil bearings rotor system

The rotordynamic performance of a new designed gas foil bearing, hybrid bump-metal mesh foil bearing (HB-MFB), was investigated in this study. The elastic substructure of the HB-MFB was comprised by the foil strips and metal mesh blocks, which is different from the traditional bump-type gas foil bearing. The metal mesh material was introduced to improve the damping of the bearing, thus improving the stability of the rotor system. The new designed HB-MFB was manufactured and tested on a rotordynamic test rig in this study. The rotor responses were measured and analyzed under different operating conditions and bearing parameters. A theoretical model of the HB-MFB rotor system was built and demonstrated by the experiment results. The performance of HB-MFB was compared with traditional bump-type gas foil bearings. The effect of bearing parameters and operating conditions, like imbalance, mesh density, bearing load, and bearing clearance, on the rotordynamic performance of HB-MFB rotor system was studied experimentally and analyzed theoretically. The experiment results demonstrated that introducing metal mesh can suppress the subsynchronous vibration significantly, which means the stability can be improved. The critical speed changes slightly, while the synchronous amplitude increases as the imbalance increases in the whole operating speed range. The rotor responses under different static loads showed the stability was improved as the static load increases because of the improvement of the natural frequency and the threshold speed of instability. The rotordynamic performance of HB-MFB rotor system is sensitive with some of the bearing parameters, like mesh density and bearing clearance.

SOM-Based High-Dimensional Design Spaces Mapping for Multi-Objective Optimization

Multi-objective optimization can reveal the complex parameter-objective relationships in the high-dimensional design problems. However, the data-extraction and data-presentation of the high-dimensional complex nonlinear system suffers from the increasing dimensionality. Key features and data-distribution of high-dimensional design spaces:parameter and objective spaces could be obtained by using Self-Organizing Maps (SOM) method, which re-clusters the high-dimensional multi-attribute data existing on the Pareto front into several low-dimensional maps. Correlations among all the design variables can be drawn according the colorized topological structure of the maps. Under the constraints including geometric structure and operating parameters, a low-cost and high accurate Kriging surrogate model was established to optimize a hybrid sliding bearing based on the sequential design method. Correlations between 3 objectives:"friction-to-load" ratio, temperature rise, instability threshold speed and 4 design parameters were extracted by SOM. Optimal feature regions were captured and analyzed. Results show that, within the specific feasible design space, supply pressure, axial bearing land width have important impact on the selected objectives, whereas the other parameters such as deep pocket depth and shallow pocket angle have relatively limited impact. A series of corresponding design decisions and optimization results help to understand the mechanism of the hybrid sliding bearing system in a much more intuitive way.

Eliminating the Fluid‐Induced Vibration and Improving the Stability of the Rotor/Seal System Using the Inerter‐Based Dynamic Vibration Absorber

A theoretical research on eliminating the instability vibration and improving the stability of the rotor/seal system using the inerter‐based dynamic vibration absorber (IDVA) is presented in this paper. The modified Jeffcott rotor and Muszynska nonlinear seal force models are employed. The proposed IDVA is the damping element of the traditional dynamic vibration absorber (DVA) replaced by one of the six configurations of the inerter. The instability threshold speed of the system is obtained by applying the Routh–Hurwitz stability criterion. The quantum particle swarm optimization (QPSO) method is utilized to optimize the parameters of the IDVA. The numerical method is applied to investigate the nonlinear dynamic responses and stability. The results show that the IDVA can effectively improve the stability and reduce the instability vibration of the rotor/seal system. Furthermore, the performance of the IDVA is more effective than that of the traditional DVA.

Stability analysis of fluid film bearings under laminar and turbulent regimes

Stability of short journal bearings is investigated in bith the linear and nonlinear regimes under laminar and turbulent operating conditions with consideration of drag force effects. Exerted forces on fluid film bearings comprises of two different components. (1) Pressure force components in radial and tangential directions, and (2) Drag force (shear/friction force) components in radial and tangential directions. In most existing literature, related to rotor bearing system supported on journal bearing stability analysis, the effect of fluid film drag force has been neglected. Utilizing Ng-Pan-Elrod model and the modified Reynold’s equation, detailed calculation of the turbulent pressure distribution is carried out. Local stability of periodic solutions was studied using Hopf bifurcation theory. The shaft stiffness and inclusion of drag force and turbulent effects were found to play an important role in stability regions and bifurcation profiles of a flexible rotor bearing system.It was found that for shafts supported on short journal bearings, consideration of drag force effect could lead to expansion of un-stable region at high Sommerfeld numbers (low shaft static loads). Dynamic coefficients of short length journal bearing were analytically calculated under laminar and turbulent regimes based on Ng–Pan–Elrod model. Linear stability charts of a flexible rotor supported on laminar and turbulent journal bearings are found by calculating the threshold speed of instability associated to the start of instable oil whirl phenomenon. Local journal trajectories of the rotor-bearing system were found at different operating conditions solely based on the calculated dynamic coefficients in laminar and turbulent flow. Results show discrepancy between laminar and turbulent models, when shaft stiffness is great than 10, for the entire range of Sommerfeld number by increasing the Reynolds number in turbulent models.

Nonlinear Dynamics of Flexible Rotors Supported on Journal Bearings—Part II: Numerical Bearing Model

The nonlinear stability of a flexible rotor-bearing system supported on finite length journal bearings is addressed. A perturbation method of the Reynolds lubrication equation is presented to calculate the bearing nonlinear dynamic coefficients, a treatment that is suitable to any bearing geometry. A mathematical model, nonlinear coefficient-based model, is proposed for the flexible rotor-bearing system for which the journal forces are represented through linear and nonlinear dynamic coefficients. The proposed model is then used for nonlinear stability analysis in the system. A shooting method is implemented to find the periodic solutions due to Hopf bifurcations. Monodromy matrix associated to the periodic solution is found at any operating point as a by-product of the shooting method. The eigenvalue analysis of the Monodromy matrix is then carried out to assess the bifurcation types and directions due to Hopf bifurcation in the system for speeds beyond the threshold speed of instability. Results show that models with finite coefficients have remarkably better agreement with experiments in identifying the boundary between bifurcation regions. Unbalance trajectories of the nonlinear system are shown to be capable of capturing sub- and super-harmonics which are absent in the linear model trajectories.

Nonlinear Dynamics of Flexible Rotors Supported on Journal Bearings—Part I: Analytical Bearing Model

To determine the bifurcation types in a rotor-bearing system, it is required to find higher order derivatives of the bearing forces with respect to journal velocity and position. As closed-form expressions for journal bearing force are not generally available, Hopf bifurcation studies of rotor-bearing systems have been limited to simple geometries and cavitation models. To solve this problem, an alternative nonlinear coefficient-based method for representing the bearing force is presented in this study. A flexible rotor-bearing system is presented for which bearing force is modeled with linear and nonlinear dynamic coefficients. The proposed nonlinear coefficient-based model was found to be successful in predicting the bifurcation types of the system as well as predicting the system dynamics and trajectories at spin speeds below and above the threshold speed of instability.

Effect of shape errors on the stability of externally pressurized air journal bearings using semi-implicit scheme

Effects of shape errors on the dynamic behavior of externally pressurized air journal bearings are studied. A semi-implicit numerical scheme is used in order to reduce the computing time. A new algorithm is developed that can handle a non-tridiagonal matrix from the periodic boundary conditions of the journal bearing. Using this method, the half frequency whirl phenomena and instability threshold speeds can be effectively calculated for the bearings having shape errors on the surfaces of the shafts and bearings, respectively. It is shown that the effect of shape errors on the stability from the shaft and the bearing is different and the precision of the shaft is much more important than that of the bearing.

Linear stability analysis of finite length journal bearings in laminar and turbulent regimes

Dynamic coefficients of a finite length journal bearing are numerically calculated under laminar and turbulent regimes based on Ng–Pan–Elrod and Constantinescu models. Linear stability charts of a flexible rotor supported on laminar and turbulent journal bearings are found by calculating the threshold speed of instability associated to the start of instable oil whirl phenomenon. Local journal trajectories of the rotor-bearing system were found at different operating conditions solely based on the calculated dynamic coefficients in laminar and turbulent flow. Results show no difference between laminar and turbulent models at low loading while significant change of the size of the stable region was observed by increasing the Reynolds number in turbulent models. Stable margins based on the laminar flow at relatively low Sommerfeld numbers [Formula: see text] were shown to fall inside the unstable region and hence rendering the laminar stability curves obsolete at high Reynolds numbers. Ng-Pan turbulent model was found to be generally more conservative and hence is recommended for rotor-bearing design.

A simpler method to calculate instability threshold speed of hydrodynamic journal bearings

In this study, a simpler method is proposed to study the instability threshold speed of journal bearing. State vector is chose as c = [ ε φε̇φ̇ ], not [ X YẊẎ ] in previous discussions, and coordinate transformation (between (ε, φ) and (X, Y)) is avoided. For the short bearing, a simpler expression is obtained without accuracy reducing and the calculation process is simplified. While for the long bearing, the instability threshold speed is expressed analytically by the eccentricity ratio εs, the attitude angle φs and the Sommerfeld number S2. The results agree well with the numerical results in previous studies. It is confirmed that, this method is more effective than the methods used in previous studies.

Rotordynamic Optimization of Fixed Pad Journal Bearings Using Response Surface Design of Experiments

The design process of journal bearings of turbomachines is complex and time-consuming due to the many geometric and physical variables involved. This paper reports on the design of experiments (DOE) and the response surface design of experiments (RSDOE) methods employed on the design of the drive-end and free-end three-lobe journal bearings supporting a centrifugal compressor rotor. The suitability of each technique is discussed. The bearing design variables employed are bearing slenderness ratio, radial clearance, preload, and lubricant inlet temperature. The rotordynamic response variables selected were the critical speed location, the vibrations at critical speed and operating speed for both bearings, and the threshold speed of instability. The use of a nonlinear (quadratic) RSDOE model is justified. An optimization approach combining an SRDOE and rotordynamic finite element modeling is presented. This method leads to arrive to a multivariate model for multi-objective optimization with very few computations. Identification of the dominant design variables and their effects on several response variables allows establishing engineering feasible solutions with focus on manufacturing versus operating conditions tradeoff.

Dynamic Stability Analysis and Critical Speed of Rotor supported by a Worn Fluid film Journal Bearings

In this paper, the effect of wear in the fluid film journal bearings on the dynamic stability of rotor bearing system has been studied depending on the development of new analytical equations for motion, instability threshold speed and steady state harmonic response for rotor with offset disc supported by worn journal bearings. Finite element method had been used for modeling the rotor bearing system. The analytical model is verified by comparing its results with that obtained numerically for a rotor supported on the short bearings. The analytical and numerical results showed good agreement with about 8.5% percentage error in the value of critical speed and about 3.5% percentage error in the value of harmonic response. The results obtained show that the wear in journal bearing decrease the instability threshold speed by 2.5% for wear depth 0.02 mm and 12.5% for wear depth 0.04 mm as well as decrease critical speed by 4.2% and steady state harmonic response amplitude by 4.3% for wear depth 0.02 mm and decrease the critical speed by 7.1% and steady state harmonic response amplitude by 13.9% for wear depth 0.04 mm.
 

Dynamic Characteristics of a Finite-Width Journal Bearing Lubricated with Powders

This paper attempts to investigate the dynamic characteristics of a powder lubricated journal bearing. The stiffness and damping coefficients are obtained using finite perturbation method. The stability limit of the rotor speed is obtained for a system consisting of a single rotor disc in the middle of a flexible shaft having identical plain cylindrical journal bearings at the ends. The threshold speed of instability for a rotor supported on powder lubricated journal bearing is compared with that of oil lubricated journal bearing. The numerical results show that a rotor supported on powder lubricated journal bearings remains stable for a speed limit much higher than that for oil lubricated bearings.