Threshold Speed Of Instability Research Articles

Optimum modeling of a flexible multi-bearing rotor system

Modeling and system identification on flexible multi-bearing rotors with two-concentrated disks are presented in this study. Both rotor unbalance vibration responses through critical speed were experimentally obtained through accurate control of journal bearing static load. Vibration simulations of this laboratory rotor-bearing system were performed as a linear system. The simulation predicted that critical speeds were higher than the experimental ones. These differences strongly occurred because of errors in estimating journal bearing damping and stiffness coefficients. Estimated journal bearing coefficients were carried out by using optimization methods to enhance the model prediction capabilities to the actual test results. The pattern search method was used to solve an inverse problem for the global parameters. A good agreement, in terms of instability threshold speeds and system responses, was found between the experimental results and the optimized model. Both the Response Surface method and the Neural Network method were used to build a Metamodel for predicting the system coefficients. For all tested static loads, the predicted error of the original model was in the range of 9% - 18%.

Effect of Different Flow Regime on the Static and Dynamic Performance Parameter of Hydrodynamic Bearing

The hydrodynamic journal bearing system has found wide spread application in high speed rotating machine such as compressors, gas turbines, water turbines, steam turbines, alternators etc. As rotor generally operates at high speed, the lubricant flow in the clearance space of journal bearing does not remain laminar and for accelerated/decelerated journals the threshold speed of instability is crossed from both sides. In this paper the numerical method has been used to compute the static and dynamic performance parameter. The analysis is carried out for the case of short bearing approximation aspect ratio (L/D under different fluid flow regime i.e. laminar, transition and turbulent flow condition assuming the perfectly rigid journal and bearing.

Experimental Study on Journal Gas Bearing with Axial Slot for a Miniature Air Turboexpander

A new type of hydrostatic journal gas bearing with axial slot of tangential gas entry for miniature air turboexpander is firstly developed in this paper. This bearing configuration is presented and constructed to fit manufacture simplicity. The experimental investigations are carried out to reveal the relationship between bearing parameters and instability threshold speed. It is feasible to increase the stability of hydrostatic journal gas bearing by adopting this kind of configuration with axial slot of tangential gas entry. Some valuable tested results are obtained to reveal the optimum slot height and radial clearance for this miniature air turboexpander.

Morton Effect Induced Synchronous Instability in Mid-Span Rotor–Bearing Systems—Part I: Mechanism Study

The Morton Effect in rotor-bearing systems may lead to an unstable operation. In Part I, the mechanism of the Morton Effect–induced thermal instability in the mid-span rotor systems is studied. First, the equivalent thermal induced imbalance is introduced and its magnitude and directions are assumed, to represent the viscous thermal effect on the rotor systems. Then, the simplified rotor and bearing models are adopted for the derivation of analytical expressions. The results show that there exists a threshold of instability due to the Morton Effect in the mid-span rotors. Based on the assumptions of linear isotopic bearing supports, this threshold speed takes a simple form, which is determined by the support stiffness and the introduced equivalent coefficient of thermal effect, for the rigid or elastic rotors, with the thermal imbalance acting in the same direction as the response displacement. The threshold of instability is also obtained for the system with the thermal imbalance acting perpendicular to the response displacement, where the supporting damping plays a role. For a perspective view of the system stability, a stability map for the damped rigid mid-span rotors with the thermal imbalance having arbitrary phase difference is generated. It shows that the stable operating regions of the system are bounded by two curves of threshold of instability, named the first and second threshold speeds of instability, respectively. The Morton Effect–induced instability thresholds are actually affected by both the magnitude and relative phase of the thermal imbalance. The mechanism of the Morton Effect–induced thermal instability of mid-span rotors supported by linear isotropic bearings can be explained through the fact that the Morton Effect introduces either negative stiffness or negative cross-coupled stiffness. In addition, the Morton Effect also has a comprehensive impact on both the amplitude and phase lag of the steady-state unbalance response. It may shift both curves in a manner dependent on the relative magnitude and direction of the thermal imbalance.

DYNAMIC MODELING OF ROTOR SHAFT WITH INTERNAL DAMPING DRIVEN THROUGH A DISSIPATIVE COUPLING

The paper presents the dynamic behavior of a hollow rotor shaft with internal damping driven by a dissipative coupling. The coupling in the system is absolutely flexible in transverse and bending; however, it is assumed to be torsionally rigid. Bond graph is adopted for modeling as it facilitates the system modeling from the physical paradigm itself, and the model can be easily extended to incorporate modifications. Simulation results show interesting phenomenon of limiting behavior of rotor shaft with internal damping beyond the threshold speed of instability. It is further shown that the shaft entrainment frequency of the rotor shaft primarily depends on the ratio of external and internal damping.

Running Characteristics of Aerodynamic Bearing with Self-Lifting Capability at Low Rotational Speed

An aerodynamic journal bearing that is capable of self-generating squeeze-film pressure is presented and its dynamic characteristics investigated numerically and experimentally. A numerical method based on a time-marching static model was applied to assess the orbit trajectory path of the rotor upon a perturbation. Experimental results were obtained to validate the effect of the self-generated squeeze-film pressure on the stability of the rotor. Analyzing the Fast Fourier Transform (FFT) responses of the rotor orbits enabled identification of self-excited whirling instabilities. Both numerical and experimental results showed that increasing the squeeze-film effect of the bearing could raise the threshold speed of instability.

Use of nonlinear journal-bearing impedance descriptions to evaluate linear analysis of the steady-state imbalance response for a rigid symmetric rotor supported by two identical finite-length hydrodynamic journal bearings at high eccentricities

This paper concerns the investigation of validity limits of linear models in predicting rotor trajectory inside the bearing clearance for a rigid symmetric rotor supported by two identical journal bearings operating at high eccentricities. The inherent nonlinearity of hydrodynamic journal bearings becomes strong for eccentricities grater than 60% of the bearing clearance where most existing linear models are not able to accurately predict the rotor trajectory. The usefulness of nonlinear journal-bearing impedance description method in this investigation is due to the analytical formulations of the linearised bearing coefficients, and the analytical nonlinear bearing models. These analytically derived bearing coefficients do not require any numerical differentiation (or integration) and are therefore more accurate for large eccentricities. The analytically derived nonlinear bearing models markedly decrease the simulation time while valid for all L/D (length to diameter ratios) and all eccentricities. The results contained in this paper show that linear models derived from the nonlinear impedance descriptions of the Moes-cavitated (π-film) finite-length bearing can predict the steady-state imbalance response of a symmetric rigid rotor supported by two identical journal bearings at high eccentricities. This is, however, only the case when operating conditions are below the threshold speed of instability and when the system has period-one solutions. The error will become larger closer to the resonance speed.

Parametric Studies on Static Performance and Nonlinear Instability of Bump-Type Foil Bearings

A theoretical analysis model is presented to investigate the nonlinear dynamic behavior of bump-type foil bearings, i.e., the instability and unbalance response. In the developed model, the foil structure of bump-type foil bearings was simulated using the link-spring model, which was presented and validated in a previous study. During the calculation, the Reynolds equation and the foil structure model were coupled though the pressure and film thickness. An iteration solution method with the equation of shaft motion was applied for the shaft orbit. Parametric studies of design parameters, such as the bump number, the length ratio of the segment between two bumps and a bump, the foil thickness, the bump height, and the Young's Modulus, were presented upon the analysis of static performance and the instability of bump-type foil bearings. It is noted that more bumps or lager length ratio lead to larger load capacity. But the bearing load capacity does not change with different bump heights. Moreover, the threshold speed of instability decreases significantly with the rise of the bump number, especially when the number of bumps is low. However, it almost does not affect by other four parameters.

Effect of Side Feed Pressurization on the Dynamic Performance of Gas Foil Bearings: A Model Anchored to Test Data

AbstractComprehensive modeling of gas foil bearings (GFBs) anchored to reliable test data will enable the widespread usage of these bearings into novel high speed turbomachinery applications. GFBs often need a forced cooling gas flow, axially fed through one end of the bearing, for adequate thermal management. This paper presents rotordynamic response measurements on a rigid rotor supported on GFBs during rotor speed run-up and coastdown tests with the GFBs supplied with increasing feed gas pressures to 2.8bars. Rotor speed run-up tests to 35krpm show that the bearing end side feed gas pressurization delays the onset speed of rotor subsynchronous whirl motions. The test data validate closely the predictions of the threshold speed of instability and the whirl frequency ratio derived from a GFB model that implements the axial evolution of gas circumferential flow velocity as a function of the imposed side feed pressure. Rotor speed coastdown tests from 25krpm with a low feed pressure of 0.35bar evidence a nearly linear synchronous rotor response for small and moderately large imbalance mass distributions. A structural finite element rotordynamics model integrates linearized synchronous speed GFB force coefficients and predicts synchronous responses, amplitude, and phase angles, agreeing with the test data. The analysis and measurements demonstrate the profound effect of the end side feed gas pressurization on the rotordynamic performance of GFBs.

Thermal effects on Farley–Buneman waves at nonzero aspect and flow angles. I. Dispersion relation

With the increasingly accepted notion that E-region electrostatic plasma waves move at their instability threshold speed when they attain their maximum amplitude, it has become more important to determine just what that threshold speed is supposed to be. An accurate determination requires a calculation of the electron temperature fluctuation level, which can sometimes dramatically influence the results. In this paper, the Grad 8-moment transport equations have been employed with the addition of a generalized electron cooling rate to account for cooling via inelastic collisions, and a general electron volume heating rate that includes frictional heating as a special case. The derived generalized dispersion relation is presented in such a way that a variety of other earlier treatments, including the isothermal and adiabatic limits and an instability associated with electron friction, all fall out transparently, allowing us to easily compare our results with earlier ones while tracking down the physical processes involved. The presentation format is also geared to facilitate the computation of the dispersion relation for the purpose of comparing the theoretical results with observations.

Effects of oil inlet pressure and inlet position of axially grooved infinitely long journal bearings. Part II: Nonlinear instability analysis

This paper extends the analysis of Part I to investigate the influence of oil inlet pressure and its position on the dynamic characteristics of axially grooved journal bearings. Using the dynamic fluid force derived in Part I, the equations of motion are solved for the locus of the journal center. A trial-and-error method is used to identify the instability threshold speed for a given Sommerfeld number S. The effects of oil inlet pressure (in the range of 0 ⩽ p ¯ i ⩽ 1 ) and inlet position (in the range of 0⩽ Θ i⩽90°) on the instability threshold speed of a rotor-bearing system are investigated and discussed. It is shown that the instability threshold speed with oil inlet position Θ i=0 decreases on increasing the oil inlet pressure p ¯ i from 0 to 1.0 while the influence of oil inlet position on the instability threshold speed depends on the steady state eccentricity ratio ε.

Experimental Response of Simple Gas Hybrid Bearings for Oil-Free Turbomachinery

Gas film bearings offer unique advantages enabling successful deployment of high-speed microturbomachinery (<0.4 MW). Current applications encompass micropower generators, air cycle machines and turbo expanders. Mechanically complex gas foil bearings are in use; however, their excessive cost and lack of calibrated predictive tools deters their application to mass-produced systems. The present investigation provides experimental results for the rotordynamic performance of a small rotor supported on simple and inexpensive hybrid gas bearings with static and dynamic force characteristics desirable in high-speed turbomachinery. These characteristics are adequate load support, stiffness and damping coefficients, low friction and wear during rotor startup and shutdown, and most importantly, enhanced rotordynamic stability. The test results evidence the paramount effect of feed pressure on early rotor lift-off and substantially higher threshold speeds of rotordynamic instability. Higher supply pressures also determine larger bearing direct stiffnesses, and thus bring an increase in the rotor-bearing system critical speed albeit with a reduction in damping ratio.

Comparison of Rotordynamic Analysis Predictions With the Test Response of Simple Gas Hybrid Bearings for Oil Free Turbomachinery

Current applications of gas film bearings in high-speed oil-free microturbomachinery (<0.4 MW) require calibrated predictive tools to successfully deploy their application to mass-produced systems, for example, oil-free turbochargers. The present investigation details the linear rotordynamic analysis of a test rotor supported on externally pressurized gas bearings. Model predictions are compared with the test rotordynamic response determined through comprehensive experiments conducted on a small rotor supported on three lobed hybrid (hydrostatic/hydrodynamic) rigid gas bearings. Predictions for the rotor-bearing system synchronous response to imbalance show good agreement with measurements during rotor coastdowns, and manifest a decrease in damping ratio as the level of external pressurization increases. The rotor-bearing eigenvalue analysis forwards natural frequencies in accordance with the measurements, and null damping ratios evidence the threshold speeds of rotordynamic instability. Estimated whirl frequency ratios are typically 50% of rotor speed, thus predicting subsynchronous instabilities at lower rotor speeds than found experimentally when increasing the magnitude of feed pressurization. Rationale asserting the nature of the discrepancies calls for further analysis.

Bifurcation Analysis of a Flexible Rotor Supported by Two Fluid-Film Journal Bearings

The effects of rotor stiffness on the bifurcation regions of a flexible rotor supported by two identical fluid-film journal bearings are presented. It is shown that the rotor stiffness has a pronounced influence on the bifurcation characteristics at the instability threshold speed. For short bearings, two bifurcation regions exist if the dimensionless rotor stiffness K¯⩾4.3. On the other hand, three bifurcation regions exist if the dimensionless rotor stiffness K¯<4.3. Information is presented that allows one to easily predict both the instability threshold speed and its bifurcation type of a rotor-bearing system with any specific set of operating parameters. The results presented have been verified by laboratory experiments as well as several published results in the open literature. Several examples are presented to illustrate the application of the results for design purposes.

Influence of Inlet Oil Temperature on the Instability Threshold of Rotor-Bearing Systems

AbstractThe existing literature contains a number of disparities in assessing the effect of inlet oil temperature on the instability threshold speed in hydrodynamic journal bearings. Specifically, some papers have presented evidence that lowering the inlet oil temperature tends to have a stabilizing effect, whereas others have shown the opposite. No clear explanation has been offered for this phenomenon. This paper presents the results of a series of experiments that explain the nature of these disparities and sheds light on the effect of the inlet oil temperature on the instability threshold in hydrodynamic journal bearings.

Stability Analysis of a Rough Journal Bearing Considering Cavitation Effects

A modified governing equation is derived incorporating the effects of roughness and cavitation in a journal bearing. The available theories of Reynolds roughness and cavitation algorithm proposed by Elrod are utilized in this work to develop a numerical procedure for stability analysis of a liquid lubricated rough journal bearing. The Elrod generalized theory of Reynolds roughness provides a governing equation based on the surface configuration. The Elrod cavitation algorithm conserves mass throughout the bearing and automatically predicts the full film and cavitation regions using a switch function. The roughness is considered on either or both the bearing and journal surfaces. The instability threshold speed increases significantly for the roughness patterns on the grooved bearing surface only at higher eccentricity ratios. The threshold speed increases significantly with increase in the inclination of herringbone type striated roughness patterns on the journal surface from 110° to 150°.

Stability analyses and experiments of spindle system using new type of slot-restricted gas journal bearings

A new type of slot-restricted gas journal bearing with non-uniform slot clearance in the circumferential direction is proposed in this paper. The stability characteristics of the new type of slot-restricted gas bearing are obtained by solving linear equations of motion of rotor with two degrees of freedom supported by one bearing to prove that the proposed new type of the slot-restricted gas journal bearing is much more stable than a conventional slot-restricted bearing. For comparisons with experimental results, the stability characteristics of four degrees of freedom rotor-bearing system are judged by the non-linear transition analyses. The instability threshold speed tests of the rotor-bearing system are performed. The instability threshold speed experimentally obtained is compared with theoretical results to confirm the validity of the numerical analyses. As a result of the analysis and experiment, it was shown that the proposed new type of slot-restricted gas journal bearing is much more stable than a conventional slot-restricted bearing due to non-uniform slot clearance.

원형 슬롯 레스트릭터를 갖는 외부 가압 공기 저널 베어링의 소음 및 진동 특성 예측

The purpose of this paper is to investigate the noise and vibration characteristics of externally pressurized air journal bearings with a circular slot restrictor. To do this, the nonlinear transient analysis including rotor imbalance was performed for a rotor-bearing system. The effects of radial clearance and the length of the bearing and mass eccentricity of the rotor on the noise and vibration characteristics of the bearing are also examined. The results show that the noise and vibration of the rotor-bearing system first increase up to critical speed of the system, and then decrease up to instability threshold speed of the system as the rotational speed of the rotor increases, and the noise of the bearing is markedly influenced by the mass eccentricity of the rotor and the radial clearance and the length of the bearing.

Fast type‐I waves in the equatorial electrojet: Evidence for nonisothermal ion‐acoustic speeds in the lower E region

There is plenty of evidence to suggest that the phase velocity of large amplitude irregularities produced by the modified two‐stream and the gradient‐drift instabilities are the same as the threshold speed, namely, the nominal ion‐acoustic speed in the modified two‐stream case. In this context, it is rather puzzling to note that the phase velocity of type‐I waves in the equatorial electrojet is known to easily exceed 400 m/s during strong electrojet conditions. This is puzzling because the ion‐acoustic speed of the medium is expected to be 100 m/s slower than these observations. Explaining the observations as an increase in the nominal ion‐acoustic speed through much higher neutral temperatures than expected or through electron heating by plasma waves is problematic at best. The first explanation violates everything we know about the neutral atmospheric temperature near the mesopause, while in the latter case, we only have to recall the emerging view that electron heating is done, at high latitudes, by parallel wave fields and that there is no evidence for the existence of such fields in the equatorial case. By contrast, we show that, contrary to what is normally assumed, electron thermal fluctuations cannot be neglected in the theoretical treatment of the instability when the ion collision frequency becomes large compared to the wave frequency and the wave aspect angle is small. These electron thermal fluctuations are caused by electron adiabatic heating and cooling effects related to the wave dynamics. When the electron thermal fluctuations are included in the calculations the derived instability threshold speeds match the upper limit reached by the observations. The increase becomes detectable at 108 km altitude and increases rapidly with decreasing altitude to become roughly 1.5 times as large as the isothermal ion‐acoustic speed below 100 km altitude. We show in this paper that the new threshold speed provided by the nonisothermal threshold calculations provides an excellent match for the largest vertical type‐I phase speeds that were observed during a very strong daytime electrojet event.

The effect of active control on stability characteristics of hydrodynamic journal bearings with an axial groove

Results of theoretical investigations on stability characteristics of an actively controlled hydrodynamic journal bearing are presented. Proportional, derivative and integral controls are adopted for a hydrodynamic journal bearing with an axial groove. Furthermore, a cavitation algorithm, implementing the Jakobsson-Floberg-Olsson boundary condition, is adopted to predict cavitation regions in a fluid film more accurately than the conventional analysis, which uses the Reynolds condition. Using the bearing's linear dynamic coefficients, which are evaluated from the perturbation method, the stability characteristics of a rotor-bearing system are investigated using the Routh-Hurwitz criteria. It is shown that the instability threshold speed of a rotor-bearing system can be greatly increased and the unbalanced responses of the system can be markedly decreased by both proportional and derivative control of the bearing. Results show that active control of a hydrodynamic journal bearing can be adopted for stability improvement and reduction of the unbalanced whirl amplitude of a rotor-bearing system.