Publications

Whirling (translational) and precession (tilt) motion of the shrouded centrifugal impeller are possible vibration sources that can cause rotordynamic instability problems. Whirling motion of shrouded impellers and seals has been investigated by test and theory in the literature. However, there has been little study of the effects of coupled motion of whirling and precession of a centrifugal impeller on rotordynamic forces and moments using computational fluid dynamics (CFD). In the present study, the CFD approach for calculating the moment coefficients of the precessing impeller is developed and verified by comparison with the measured data for a precessing centrifugal compressor by Yoshida et al. (1996, “Measurement of the Flow in the Backshroud/Casing Clearance of a Precessing Centrifugal Impeller,” Sixth International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Honolulu, Hawaii, Vol. 2, pp. 151–160). A full set (4 × 4) of rotordynamic coefficient matrices is calculated, using two separate models of (a) a precessing impeller with a tilt angle and (b) a whirling impeller with dynamic eccentricity to investigate the stability of the impeller. Rotordynamic stability is evaluated by using the whirl frequency ratio of the coupled motion, obtained from the full rotordynamic coefficient matrices, to show that the precession motion has a significant impact on rotordynamic stability. A similar conclusion is reached based on the whirling plus precession response of a finite element (FE) structural rotordynamic model including the 4 × 4 rotordynamic coefficient matrices. A stability analysis using the rotordynamic coefficients indicates that the precession motion with the positive tilt angle increases the tendency toward destabilization of the rotor.

The authors present an improved modeling approach to analyze the coupled rotor-support dynamics by modeling the rotor with solid finite elements (FEs) and utilizing multiple-input and multiple-output transfer functions (TFs) to represent the flexible support. A state-space model is then employed to perform general rotordynamic analyses. Transfer functions are used to simulate dynamic characteristics of the support structure, including cross-coupling between degrees-of-freedom. These TFs are derived by curve-fitting the frequency response functions of the support model at bearing locations. The impact of the polynomial degree of the TF on the response analysis is discussed, and a general rule is proposed to select an adequate polynomial degree. To validate the proposed approach, a comprehensive comparison between the complete solid FE rotor-support model (CSRSM) and the reduced state-space model (RSSM) is presented. Comparisons are made between natural frequencies, critical speeds, unbalance response, logarithmic decrement, and computation time. The results show that the RSSM provides a dynamically accurate approximation of the solid FE model in terms of rotordynamic analyses. Moreover, the computation time for the RSSM is reduced to less than 20% of the time required for the CSRSM. In addition, the modes up to 100,000 cpm are compared among the super-element, beam element, and RSSM. The results show that the RSSM is more accurate in predicting high-frequency modes than the other two approaches. Further, the proposed RSSM is useful for applications in vibration control and active magnetic bearing systems.

 

Solid Element Rotordynamic Modeling of a Rotor on a Flexible Support Structure Utilizing Solid Element Modeling and Support Transfer Functions

This paper performs the parametric studies corresponding with the theoretical Morton effect (ME) model explained in Part I of this paper, where the fully nonlinear transient analysis based on the finite element method is introduced. Operating parameters, such as oil supply temperature, bearing clearance, oil viscosity, etc., are perturbed from the testing conditions to investigate the shifting of critical speeds and ME instability onset speed (IOS). The ME is significantly affected by the rotor bending mode with large overhung deflections, and operating parameters should be adjusted to increase the separation margin between the operating speed and the corresponding critical speed for ME mitigation. Reducing the carryover flow ratio and using the asymmetric bearing pivot offset are capable to suppress the ME by reducing both the average and differential journal temperature. The heat barrier sleeve with air or ceramic isolation is designed to prevent the heat flux into the journal and can successfully mitigate the ME based on the simulations.

 

Double Overhung Disk and Parameter Effect on Rotordynamic Synchronous Instability-Morton Effect Part II Occurrence and Prevention

The Morton effect (ME) is a thermally induced instability problem that most commonly appears in rotating shafts with large overhung masses, outboard of the bearing span. The time-varying thermal bow due to the asymmetric journal temperature distribution may cause intolerable synchronous vibrations that exhibit a hysteresis behavior with respect to rotor speed. The fully nonlinear transient method designed for the ME prediction, in general, overhung rotors is proposed with the capability to perform the thermoelastohydrodynamic analysis for all the bearings and model the rotor thermal bow at both overhung ends with equivalent distributed unbalances. The more accurate nonlinear, coupled, double overhung approach is shown to provide significantly different response prediction relative to the more approximate linear method based using bearing coefficients and the single-overhung method, which assumes that the ME on both rotor ends can be decoupled. The flexibility of the bearing pad and pivot is investigated to demonstrate that the pivot flexibility can significantly affect the rotordynamics and ME, while the rigid pad model is generally a good approximation.

 

Double Overhung Disk and Parameter Effect on Rotordynamic Synchronous Instability-Morton Effect Part I Theory and Modeling

This paper introduces a numerical scheme for simulating instabilities of a nonlinear rotordynamic system including thermal effects in the fluid film bearings. The method utilizes shooting/arc-length continuation, and simultaneous, finite element based solutions of the variable viscosity Reynolds equation and the energy equation. This provides a means to investigate the effects of the thermo-hydrodynamic THD model on bifurcations and nonlinear rotordynamic stability. A “Jeffcott” type rigid rotor is modeled as supported on double-layered fluid film, floating ring bearings (FRB). The FRB are known to produce highly nonlinear forces as functions of relative and absolute internal displacements and velocities. Both autonomous (free vibration) and non-autonomous (mass unbalanced excitation) cases and algorithms are presented. The computational workload and execution time required for determining coexisting periodic solutions is significantly reduced by employing deflation and parallel computing methods. The THD model nonlinear responses and bifurcation diagrams are compared with isoviscous model results for various lubricant supply temperatures. The autonomous case, THD model orbit sizes and onset of Hopf and saddle–node bifurcations for coexisting steady state responses, may have significant differences relative to the isothermal model results. The onset of Hopf bifurcation is strongly dependent on thermal conditions, and the saddle–node bifurcation points are significantly shifted compared to the isothermal model. This tends to increase the likelihood of bifurcation from a machine operators standpoint. In the non-autonomous case, large unbalance forces create sub-synchronous and quasi-periodic responses at low spin speeds. The responses stability and onset of bifurcations of these responses are highly reliant on the lubricant supply temperature.

 

Effects of Thermo hydrodynamic (THD) Floating Ring Bearing Model on Rotordynamic Bifurcation

The Morton effect (ME) is a thermally induced instability problem that most commonly appears in rotating shafts with large overhung masses and supported by fluid-film bearings. The time-varying thermal bow, due to the asymmetric journal temperature distribution, may cause intolerable synchronous vibrations that exhibit a hysteresis behavior with respect to rotor speed. First discovered by Morton in the 1970s and theoretically analyzed by Keogh and Morton in the 1990s, the ME is still not fully understood by industry and academia experts. Traditional rotordynamic analysis generally fails to predict the potential existence of ME-induced instability in the design stage or troubleshooting process, and the induced excessive rotor vibrations cannot be effectively suppressed through conventional balancing, due to the continuous fluctuation of vibration amplitude and phase angle. In recent years, a fast growing number of case studies of ME have sparked academic interest in analyzing the causes and solutions of ME, and engineers have moved from an initial trial and error approach to more research inspired modification of the rotor and bearing. To facilitate the understanding of ME, the current review is intended to give the most comprehensive summary of ME in terms of symptoms, causes, prediction theories, and solutions. Published case studies in the past are also analyzed for ME diagnosis based on both the conventional view of critical speed, separation margin (SM), and the more recent view of the rotor thermal bow and instability speed band shifting. Although no universal solutions of ME are reported academically and industrially, recommendations to help avoid the ME are proposed based on both theoretical predictions and case studies.

 

A Review of the Rotordynamic Thermally Induced Synchronous Instability (Morton) Effect

The subject of this work is the development of a novel vibration assisted seawater desalination approach which changes thermodynamic (mass transfer coefficient) and hydrodynamic (cross-flow velocity) properties to mitigate the inorganic fouling for RO membranes. A classical mass transport model and experimental measurements showed that an increased cross-flow velocity in the feed channel enhances the near membrane mass transfer coefficient, which promotes the back-diffusion of inorganic salts and reduces the concentration polarization (CP). Then a theoretical CP Finite Element Method (FEM) model incorporating increased cross-flow velocity reveals that a lower CP modulus forms near the membrane surface with a higher vibration frequency, which results in less fouling on the membrane surface. The vibration assisted desalination process was demonstrated using a linear motor driven, periodically oscillating desalination cell. A smaller flux decline was observed while using a higher vibration frequency (with a constant sinusoidal amplitude) and a higher vibration velocity (in multiple vibration forms). Process simulations and experimental observations validated that the proposed vibration assisted desalination process helps enhance the permeate flux and mitigate the formation of inorganic fouling on the RO membrane surface.

 

Reverse Osmosis Membrane, Seawater Desalination with Vibration Assisted Reduced Inorganic Fouling

This paper presents a new approach for modeling flexure-pivot journal bearings (FPJB) employing a three-dimensional (3D) elasto-hydro-dynamic (EHD) lubrication model. The finite element (FE) method is adopted for an analysis of the (1) pad-pivot dynamic behavior and the (2) fluid force. The isoviscosity Reynolds equation is utilized to calculate the fluid force acting on a flexure-pivot pad bearing and spinning journal. Computational efficiency is achieved utilizing modal coordinate transformation for the flexible pad-pivot dynamic analysis. Fluid film thickness plays a critical role in the solution of Reynolds equation and is evaluated on a node-by-node basis accounting for the pad and web deflections. The increased fidelity of the novel modeling approach provides rotating machinery designers with a more effective tool to analyze and predict rotor–bearing dynamic behavior.

 

Numerical Modeling and Analysis of Flexure-Pivot Tilting-Pad Bearing

Nonlinear vibrations and their control are critical in improving the magnetic bearings system performance and in the more widely spread use of magnetic bearings system. Multiple objective genetic algorithms (MOGAs) simultaneously optimize a vibration control law and geometrical features of a set of nonlinear magnetic bearings supporting a generic flexible, spinning shaft. The objectives include minimization of the actuator mass, minimization of the power loss, and maximization of the external static load capacity of the rotor. Levitation of the spinning rotor and the nonlinear vibration amplitude by rotor unbalance are constraint conditions according to International Organization for Standardization (ISO) specified standards for the control law search. The finite element method (FEM) was applied to determine the temperature distribution and identify the hot spot of the actuator during steady-state operation. Nonlinearities include magnetic flux saturation, and current and voltage limits of power amplifiers. Pareto frontiers were applied to identify and visualize the best-compromised solutions, which give a most compact design with minimum power loss whose vibration amplitudes satisfy ISO standards.

 

Multi-Objective Optimization Design of Nonlinear Magnetic Bearing Rotordynamic System

The double-sided fluid film force on the inner and outer ring surfaces of a floating ring bearing (FRB) creates strong nonlinear response characteristics such as coexistence of multiple orbits, Hopf bifurcation, Neimark-Sacker (N-S) bifurcation, and chaos in operations. An improved autonomous shooting with deflation algorithm is applied to a rigid rotor supported by FRBs for numerically analyzing its nonlinear behavior. The method enhances computation efficiency by avoiding previously found solutions in the numerical-based search. The solution manifold for phase state and period is obtained using arc-length continuation. It was determined that the FRB-rotor system has multiple response states near Hopf and N-S bifurcation points, and the bifurcation scenario depends on the ratio of floating ring length and diameter (L/D). Since multiple responses coexist under the same operating conditions, simulation of jumps between two stable limit cycles from potential disturbance such as sudden base excitation is demonstrated. In addition, this paper investigates chaotic motions in the FRB-rotor system, utilizing four different approaches, strange attractor, Lyapunov exponent, frequency spectrum, and bifurcation diagram. A numerical case study for quenching the large amplitude motion by adding unbalance force is provided and the result shows synchronization, i.e., subsynchronous frequency components are suppressed. In this research, the fluid film forces on the FRB are determined by applying the finite element method while prior work has utilized a short bearing approximation. Simulation response comparisons between the short bearing and finite bearing models are discussed.

 

Shooting with Deflation Algorithm Based Nonlinear Response and Neimark-Sacker Bifurcation and Chaos in Floating Ring Bearing System