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In the paper a statistical distribution of reaction forces in a vibrating rotor is traced back to assigned tolerances on components making up this rotor. First, the statistics of the unbalanced mass in an n-ring assembly is considered and an explicit expression for it is derived in terms of tolerances assigned to individual rings. As an illustrative example, the problem of vibrations of a rotor with two assembled disks is considered. The disks are manufactured to different specifications and the statistical distributions of reaction forces are found by Monte Carlo simulation. The latter allows to estimate the probability of vibrations or forces exceeding a specified level for a given accuracy of manufacturing. The methodology presented establishes a direct link between the manufacturing specifications for the components and the quality of performance of the equipment. [DOI: 10.1115/1.1480021]

This paper introduced a new approach that employed the assumed-modes method and the receptance method, to the sensitivity analysis and the optimization of rotor-bearing systems. First, the frequency equation in terms of receptances was derived. The natural frequencies and the critical speeds for a typical rotor system were then illustrated. Beginning with the receptance equation, the authors, for the first time, derived a sensitivity matrix and employed it into an optimization process. The topographical method in conjunction with the variable metric method followed for the optimal solution. In the solution process, the sensitivity matrix provided important information for search direction. Examples of critical speeds adjustment via supports change in an optimal sense were illustrated. Numerical results showed that the approach was very efficient and the solutions were very accurate. This approach, in addition, provided such valuable information as which supports dominated specific critical speeds. The developed approach proved to be very helpful to rotor engineers in both rotor modification and rotor design. [DOI: 10.1115/1.1456083]

The hybrid squeeze-film damper bearing with active control is proposed in this paper. The pressure distribution and the dynamics of a rigid rotor supported by such bearing are studied. A PD (proportional-plus-derivative) controller is used to stabilize the rotor-bearing system. Numerical results show that, due to the nonlinear factors of oil film force, the trajectory of the rotor demonstrates a complex dynamics with rotational speed ratio s. Poincare maps, bifurcation diagrams, and power spectra are used to analyze the behavior of the rotor trajectory in the horizontal and vertical directions under different operating conditions. The maximum Lyapunov exponent and fractal dimension concepts are used to determine if the system is in a state of chaotic motion. Numerical results show that the maximum Lyapunov exponent of this system is positive and the dimension of the rotor trajectory is fractal at the nondimensional speed ratio s = 3.0, which indicate that the rotor trajectory is chaotic under such operation condition. In order to avoid the nonsynchronous chaotic vibrations, an increased proportional gain is applied to control this system. It is shown that the rotor trajectory will leave chaotic motion to periodic motion in the steady state under control action. [DOI: 10.1115/1.1448318]

A technique is introduced to achieve transient vibration attenuation in a multi-input, multi-output flexible rotor/magnetic bearing system. The strategy employs feedback control of measured harmonic components of rotor vibration. Whereas previous harmonic controllers have been based only on steady state vibration characteristics, the new controller also incorporates the transient dynamics. The controller may still be designed from measured data and is determined from target transient vibrational responses arising from step changes in particular disturbances. Account is taken of delays arising from evaluation of harmonic components. Furthermore, stability boundaries for the controller are shown to have significant tolerance to measurement error. The controller is validated experimentally in a flexible rotor/magnetic bearing system and mass loss tests are used to demonstrate rapid decrease in vibration levels with near elimination of transient overshoot. [DOI: 10.1115/1.1448321]

System instability and chaotic response are the failure modes that could significantly impact the reliability and operating safety of high-speed rotor-dynamical machines. Initiation and propagation of surface cracks in rotary shafts are common causes for such failure modes. To be able to detect the onset and progression of these faults will considerably extend the lifetime and improve the reliability of the mechanical system. A waveletbased algorithm effective in identifying mechanical chaotic response has been applied to determine the nonlinear dynamical characteristics of a model-based, cracked rotor. This investigation confirms reported correlation of surface crack breathing with rotor chaotic motions. The effectiveness of the algorithm in detecting rotor-dynamic instability induced by mechanical faults as contrast to algorithms that are based on nonlinear dynamics is discussed. The results show not just the feasibility of the algorithm in mechanical fault diagnosis but also suggest its applicability to in-line, real-time condition monitoring at both the system and component levels. [DOI: 10.1115/1.1421053]

Results are presented of an experimental investigation into the aeroacoustic response characteristics of rotor turbulence ingestion. To fully characterize the rotor response, both the ingested velocity field and the resulting far-field sound were measured. The results are presented of a detailed velocity characterization, which was performed upstream of the rotor. The velocity measurements included an evaluation of the streamwise development of turbulence characteristics downstream of the grid, a high-resolution mapping of the spatial distribution of the mean velocity and rms turbulence fluctuations in the rotor inlet plane, and the development of a semi-empirical, functional representation of the three-dimensional wave number spectral density of the ingested turbulence. These data comprise a comprehensive empirical velocity model that was developed for specific application to rotor turbulence ingestion noise (Wojno, J. P., Mueller, T. J., and Blake, W. K., 'Turbulence Ingestion Noise, Part 2: Rotor Aeroacoustic Response to Grid-Generated Turbulence,' AIAA Journal, Vol. 40, No. 1, 2002, pp. 26-32).

The permanent-magnet motor is often the most important element in hard disk drive (HDD) spindles and also a frequent source of vibration and acoustic noise. The eccentricity between stator and rotor is inevitably introduced during the manufacturing process, and includes mass unbalance, shaft bow, and bearing tolerances. This paper analytically discusses the effects of rotor eccentricity on motor performance for symmetric and asymmetric motors, such as local traction, unbalanced force, cogging torque, back electromotive force (EMF), phase current, and torque ripple. An asymmetric motor, mostly chosen to reduce the cogging torque, shows a worse effect on cogging torque and unbalanced force when the eccentricity exists. It also adversely affects the flux linkage and introduces variation of the phase back EMF depending on winding and rotating eccentricity, thus yielding increased mutual torque ripple. Index Terms--Cogging force, hard disk drive spindle motor, rotor eccentricity, torque ripple, unbalanced radial forces.

Research describing the vibration of the rotor in a high temperature superconducting permanent magnetic bearing system is presented. Laser displacement sensors are used to measure the mechanical resonant state of the rotor vibration.

The higher-order effect of the stator leading-edge suction on radiated sound reveals that the suction force at the leading-edge vicinity oscillates at twice the frequency of the blade loading with a reasonable floor level for sound generation.

This work examines the free vibration and stability of a spinning, elastic disk-spindle system. The extended operator formulation is exploited to discretize the system using Galerkin's method (Parker, 1999). The coupled vibration modes of the system consist of disk modes, in which the disk dominates the system deformation, and spindle modes, in which the spindle dominates the system deformation. Both the natural frequencies and vibration modes are strongly affected by disk flexibility. If the membrane stresses associated with disk rotation are neglected then the system exhibits flutter instabilities, but these instabilities are not present when membrane stresses are modeled. Natural frequency veering between disk and spindle frequencies is prominent at low speeds and substantially affects the spectrum and stability. No veering is observed at high speeds where rotational stress stiffening diminishes disk-spindle coupling and causes the natural frequencies to converge to those of a rotating spindle carrying a rigid disk. Changes to the vibration modes are examined in terms of a strain energy ratio measuring the contribution of the disk strain energy to the total modal strain energy.

An extensive hydro-generator rotordynamic model is presented along with a technique for determining imbalances and shaft misalignments using the model and mechanical run vibration measurements. The hydro-generator model combines a finite element rotor model with a finite difference guide bearing model, generator magnetic forces and turbine hydraulic forces to calculate natural frequencies, stability and steady-state response. The complete model allows a great number of inputs including imbalances, rotation speed, guide bearing misalignments, coupling misalignments, bearing clearances, bearing temperatures, generator stator and rotor center misalignments, generator average air gap, generator power output and turbine blade tip clearance, all of which are described in this paper. Finally, the numerical results are compared with vibration data from the mechanical run and load test of an actual hydro generator unit.

The rotordynamic damping due to aerodynamic forces in a single-stage unshrouded aero turbine with a whirling rotor has been studied. The research uses the analytical fluid dynamic model developed by Martinez-Sanchez and Song. It uses harmonic analysis to find the concentric and eccentric solutions. The role in damping of turbine design parameters, such as the design flow coefficient and the axial gap between stator and rotor blades, has also been investigated.

In statically indeterminate rotor bearings systems, where the rotor is supported by one or more hydrodynamic bearings, the reactions at each hydrodynamic bearing, and hence its stiffness and damping properties depend not only on the bearing type, the operating conditions and the bearing dimensions but also on the relative lateral alignment between the journal and the bearing housing; the alignment, therefore, has a significant influence on the system stability and unbalance response. Additional complications arise if nonsymmetric bearing types such as elliptic or tilting pad bearings are present. An iterative procedure is outlined which enables the bearing reactions to be determined at any speed, thereby enabling even large systems such as turbomachinery to be rapidly analyzed in conjunction with existing linear rotor bearing vibration analysis software. Sample numerical examples show how misalignment and bearing type can affect the natural frequencies, the stability threshold, and the unbalance response of such statically indeterminate systems.

Rotors with three or more fluid-film bearings (or fluid seals) have 'redundant' supports, and, therefore, interdependent bearing loads that are generally unknown both in magnitude and direction. The steady-state bearing eccentricities and the dynamic stiffness and damping coefficients of the bearings are, therefore, also unknown since both are functions of the bearing loads. Thus, the dynamic behavior of multibearing rotors generally cannot be predicted with good accuracy without access to a procedure for calculating the steady-state bearing loads and eccentricities. This paper outlines such a procedure in terms of both the influence coefficient method, the transfer matrix method, and the finite element method. Radial bearing misalignment and flexibility of the bearing back-up structures are accounted for. Once the eccentricities are available, the bearing stiffness and damping coefficients can be calculated in the usual way and used to predict critical speeds, instability threshold speed, and rotor response to imbalance. A numerical example is presented that illustrates some of the nonlinear effects of bearing support redundancy, notably the large variations in instability threshold speed with radial bearing misalignment. The example shows how the method can be used to determine the level of bearing misalignment that leads to optimum rotor stability. It is concluded that no simple guide lines exist by which optimum stability can be achieved. Neither perfect bearing alignment nor equal load sharing between bearings necessarily lead to optimum stability.

A method is presented for reducing the lateral response of an imbalanced rotor accelerating or decelerating through its first lateral bending critical speed by using a variable acceleration rate. A lumped parameter model along with a numerical integration scheme is used to simulate the response of a simply supported, single disk rotor during fast acceleration and deceleration through critical speed. The results indicate that the maximum response and/or the total vibrational energy of a rotor passing through the critical speed can be reduced significantly by using a variable acceleration schedule. That is, reducing the acceleration rate after the nominal critical speed is passed. These predictions were verified experimentally for a single disk rotor.

An experiment was conducted to validate the lifting surface theory which stresses the rotor-stator interaction noise. A through-flow nacelle with a hub, a bladed rotor and an 18-vane stator were utilized in the experiment. Results revealed that sound was produced by the interaction of rotor viscous wakes with the downstream stator vanes as shown in the rotor-stator configurations. The results of the experiment seemed to validate the lifting surface theory.

The objective of this paper is to describe an imbalance compensation scheme for a rigid rotor supported by magnetic bearings that performs on-line identification of rotor imbalance and allows imbalance cancellation under varying speed of rotation. The proposed approach supplements existing magnetic bearing controls which are assumed to achieve elastic suspension of the rotor. By adopting a physical model of imbalance and utilizing measurements of the spin rate, the proposed algorithm allows the computation of the necessary corrective forces regardless of variations in the spin rate. Convergence of the algorithm is analyzed for single-plane balancing, and is supported by simulation in single- and two-plane balancing, as well as by experimental results in single-plane implementation.

This paper deals with a nonstationary vibration of a rotor due to its collision with a guard during passage through a critical speed. An unbalanced rigid rotor supported by springs and dampers is accelerated at a constant angular acceleration and collides with an annular guard supported by springs and dampers. This dynamic process is calculated by the Runge-Kutta method, and effects of system-parameters on the process are discussed. The collision phenomenon is analyzed through two different theories. In the collision theory, the law of conservation of momentum and the coefficient of restitution are used in order to obtain rotor and guard velocities after collision. The impulse of the force induced by collision is assumed to be equal to the momentum change before and after collision. In the contact force theory, the contact force is assumed to be proportional to the overlapped displacement of the two bodies. Few differences are observed between the calculated responses based on the two theories. In some cases, the rotor executes a diverging backward whirl due to the friction force that occurs during collision with the guard and can not pass through the critical speed. The criteria maps for nonoccurrence of the backward whirl are shown.

An innovative piezoelectric motor with a stator which is made out of piezomembrane can be utilized effectively in analogic instrumentation applications due to its thickness, torque and high speed. The ultrasonic motor, which may be used as substitute to moving-coil, includes a stator, rotor and a base. The motor performs dependently on the rotor weight and diameter and it offers enhanced operating frequency.

The vibrations of the in-plane rotor system with nonlinear elastic and geometric properties are considered in this paper. The vibrations of the rotor are described as the system of ten partial differential equations and ten boundary conditions. Approximate Galerkin procedure is used to obtain the second order nonlinear differential equation. The response frequencies of the system are discussed. It is shown that they are functions of angular velocity of the rotor and nonlinear properties of the system.

This paper investigates the unbalance response of a squeeze film damper-mounted flexible rotor. Both concentric and eccentric damper operations are examined. Comparison between predicted and experimental results shows that the 'O' film model is preferred to the 'Pi' film model. Near the first pin-pin critical speed the 'O' film model underestimates the damping effect and overestimates the centralizing force of the damper. By choosing a lower film rupture pressure in the theoretical model, better matches to the experimental results are obtained. The effects of air entrainment are shown to produce unsteady responses and jump phenomena of softening spring type. It is concluded that damper models having constant film rupture pressures do not truly represent the cavitation problem in squeeze film dampers.

An assumed modes method is developed to investigate the effect of multiple flexible disks on the vibrational modes of a flexible rotor system. The analysis rotor model consists of multiple disks on a flexible shaft, as used for steam turbines or computer storage devices. In modeling the system, centrifugal stiffening and disk flexibility effects are taken into account. To demonstrate the effectiveness of the method, a hard disk drive spindle system commonly used in personal computers and a simple flexible rotor system with two disks are selected as examples. In particular, the dynamic coupling between the vibrational modes of the shaft and the duplicate disks is investigated with the shaft rotational speed varied.

The existence of a 'breathing' crack can not only cause variations in harmonic components but also produce extra transient components into vibration signals of a rotor system. In the present paper, the composition of cracked rotor vibration signals is studied first by some analytical methods. based on this study, an approach of signal processing is suggested to detect transient components of the vibration signals and thereby the crack.

Hamilton's principle was employed in developing coupled governing differential equations for solving coupled bending-bending vibration in elastically-restrained nonuniform beams with arbitrary pretwist. The general system's field transfer matrix was expressed in integration form. The exact matrix is derived when matrix elements are analytically integrated. Results show the periodic influence of the pretwist angle on natural frequencies.

This paper describes the proportional integral (PI) control of hybrid squeeze film dampers (HSFDs) for active control of rotor vibrations. Recently it was shown that the automatically controlled HSFD based on feedback of rotor speed can be a very efficient device for active control of rotor vibration when passing through critical speeds. Although considerable effort has been put into the study of steady-state vibration control, there are few methods in the literature applicable to transient vibration control of rotor-bearing systems. Rotating machinery may experience dangerously high dynamic loading due to the sudden mass unbalance that could be associated with blade loss. Transient run-up and coast down through critical speeds when starting up or shutting down rotating machinery induces excessive bearing loads at criticals. In this paper, PI control is proposed as a regulator for the HSFD system to attenuate transient vibration for both sudden unbalance and transient run-up through critical speeds. A complete mathematical model of this closed-loop system is simulated on a digital computer. Results show an overall enhanced behavior for the closed-loop rotor system. Gain scheduling of both the integral gain and the reference input is incorporated into the closed-loop system with the PI regulator and results in an enhanced behavior of the controlled system.

Recent experimental results have demonstrated the effectiveness of adaptive open-loop control algorithms for the suppression of unbalance response on rotors supported in active magnetic bearings. Herein, tools for the analysis of stability and performance robustness of this algorithm with respect to structured uncertainty are derived. The stability and performance robustness analysis problems are shown to be readily solved using a novel application of structured singular values. An example problem is presented which demonstrate the efficacy of this approach in obtaining tight bounds on stability margin and worst case performance.

An effective technique is applied to the suppression of vibrations in flexible rotor-bearing systems with small gyroscopic effects. A balanced linear-quadratic-Gaussian (LQG) controller design procedure is implemented. The size of the controller is reduced in two stages by using (i) a balanced model reduction, and (ii) an LQG balanced reduction. The condition for a gyroscopic matrix is developed that allows one to ignore the rotor gyroscopic effects in the process of the controller design, although they are included in the rotor dynamics. The approach is illustrated on a typical rotor-bearing system represented by a 48 degree-of-freedom finite element model.

The nonlinear vibrations of a rotordynamic system that is subjected to nonsymmetric bearing clearance effects are considered. Particular attention is paid to the nonstationary response that occurs during passage through resonance. Based on an extended Jeffcott rotor model that includes the effects of nonsymmetric bearing clearances, the steady-state oscillations and their stability are also analyzed. The 'normal-loose' rotor stiffness condition that occurs due to the bearing clearance effects is modeled through a discontinuous radial stiffness model. Expressions for the amplitude and phase-modulating functions are obtained. Frequency-amplitude relationships are derived. The effects of 'normal-loose' rotor stiffness condition on the modulation of both the amplitude and phase of the near-resonance vibrations, as well as the stability of the rotor system, are systematically brought out. The effects of high-speed acceleration during run-up of the rotor are both qualified and quantified. Similarly, the case of run-down is analyzed.

A rotor system with two orthogonal lateral and two angular (torsional) degrees of freedom is considered. The rotor has asymmetry of the lateral stiffness and is laterally loaded with a constant radial force and a rotating unbalance. Constant driving and load torques are applied to the rotor. The important part of the research includes an analysis of 'snapping' action, when, during rotation, the rotor experiences a peak of torsional acceleration. This occurs when the 'strong stiffness' axis of the anisotropic rotor passes under the axis of the sideload. The numerical simulation of the analytical model exhibits a snapping (accelerated) torsional response of the rotor at twice synchronous frequency (2x), and it is especially pronounced at 1x and 2x torsional resonances. The snapping response can initiate a rotor crack in the area of stress concentration, can stimulate existing crack propagation, and can be a cause of the coupling failure. The analytical results are obtained by the Averaging Method application. They confirm the numerical results and show the possibility of combination resonance occurrences. The synchronous dynamic stiffness for the frequency range around 1x lateral resonance is analytically obtained. The specific shape of the quadrature dynamic stiffness component can serve as a shaft crack indicator and can be used for early detection of a lateral crack on the rotor.

Experimental determination of both steady-state and dynamic performance of a journal bearing requires the use of a high precision spindle with a vanishingly small range of run-out. This was achieved by first eliminating the mechanical run-out of the spindle by grinding the journal specimen while rotating in place. Once the mechanical run-out was removed, the electrical run-out sensed by the displacement proximity-probe-transducers was also removed. Using this procedure the mechanical and electrical run-outs of a research spindle were reduced to less than 0.2 micron (10 [micro]in.), which is better than the resolution of the data acquisition system, 1 micron (50 [micro]in.).

Stability of a rotor system induced by Coulomb friction in joints is investigated both theoretically and experimentally. The vibrating system consists of two shafts supported flexibly and connected by three universal joints including a cross-groove (CG) joint. The Coulomb friction in a CG joint, which is one of the most destabilizing joints among practically used ones, is treated. The friction is assumed to be a function of the relative angular velocity between the connected shafts. Three stabilizing effects are investigated: the initial intersecting angle in the CG joint, the asymmetry of the support stiffness and a dynamic damper (D/D). As a result, it is found that the initial intersecting angle has a remarkable stabilizing effect; D/D expands the stable region of the system with an initial intersecting angle; the asymmetry of the support stiffness also has a stabilizing effect on the system with an initial intersecting angle. These stabilizing effects are experimentally confirmed.

A new design methodology for the vibration control of rotor systems with a magnetic bearing is developed in this paper. The methodology combines the experimental design method in quality control engineering and the conventional PD control technique such that their advantages in implementation feasibility and performance-robustness can be integrated together. A quality loss index defined by the summation of the infinity norm of unbalanced vibration is used to characterize the system dynamics. By using the location of the magnetic bearing and PD feedback gains as design parameters, the controller can be determined by a small number of matrix experiments to achieve the best system performance. In addition, it is robust to the vibration modes within a desired speed range. A rotor system consisting of 4 rigid disks, 3 isotropic bearings, and 1 magnetic bearing is applied to illustrate the feasibility and effectiveness of the experiment-aided controller design.

In this paper a digital numerical simulation is carried out in which active magnetic bearings (AMB) are applied to control nonlinear and nonsynchronous vibrations of a rigid rotor excited by nonconservative cross-coupling mechanisms. Through an adaptive control algorithm of the AMB controller, unknown cross-coupling parameters of a rotor are estimated on-line by a standard least-square estimator along with a time-varying so-called forgetting factor. The parameters of the AMB controller are adapted in order to compensate for the cross-coupling effects and to stabilize the rotor system. Transient cross-coupling response characteristics are calculated by numerical simulation of a rigid rotor supported by two active magnetic bearings. With this kind of control strategy, the stability of the system can be guaranteed for much higher values of cross-coupling coefficients than with common nonadaptive feedback controllers, designed with pole placement or least square algorithms.

The paper describes a synchronous vibration instability problem encountered on a centrifugal compressor with oil-lubricated bearings. The problem was solved by modification of the compressor rotor; however, the root cause was not completely understood at that time. A possible explanation was based on a theory that suggested differential heating of the bearing journals. It was decided to verify this theory by experiments. Therefore a test rotor was designed with identical rotor dynamic characteristics to those of the compressor rotor. To fill a gap in the published research on bearing thermohydrodynamics, an experimental technique was devised to measure the surface temperature variations around one of the journals of this rotor. The dependence of significant temperature differentials across the journal upon its orbit was confirmed.

Squeeze film dampers (SFD) provide load isolation and attenuate rotor vibrations in high speed turbomachinery. Operating parameters such as whirl frequency, amplitude of journal motion, and value of external pressure supply determine the SFD dynamic force response and its dissipation of mechanical energy. Measurements of pressure fields and fluid film forces in a fully submerged open-end squeeze film damper are presented for tests with rotor speeds to 5000 cpm and low supply pressures. The damper has a clearance of 381 [[micro]meter] (0.015 in.) and the journal describes circular centered orbits of amplitudes ranging from 30 to 50 percent of the bearing clearance. Experimental film pressures depict a vapor cavitation (close to zero absolute pressure) zone increasing in extent as the whirl frequency increases. Estimated fluid film forces from the measured pressure profiles are found to be proportional to whirl speed and lubricant viscosity. Test cross-coupled damping coefficients ([C.sub.rt]) are smaller than predicted values based on the short-length bearing model with a [Pi] film cavitation assumption. The direct damping coefficients ([C.sub.tt]) are larger than theoretical values, especially at low frequencies where the dynamic cavitation region has not grown to half the circumferential flow extent. The experiments demonstrate the viscous character of the fluid film forces in a SFD test apparatus where fluid inertia effects are minimal (squeeze film Reynolds number less than one). On the other hand, the extent of the cavitation zone appears to be dominant on the generation of fluid film forces.

A study is presented in which the vibrations of a rotor are controlled under both synchronous and transient conditions. A Fourier decomposition of the rotor vibration during successive synchronous cycles is performed and the fundamental amplitude component is monitored. An analysis of the component response is developed and a strategy is proposed for the rotor vibration control. This is based around a gain matrix derived for a steady-state open-loop controller and the loop is then closed in a form of generalized integral control using a single feedback parameter. Performance and stability characteristics are investigated and for a linear system it is shown that the control of transient vibration caused by sudden mass loss from a rotor is possible. Limitations are stated if the system becomes nonlinear. Multiprocessor hardware is used for on-line implementation of the controller algorithm to an experimental rig, consisting of a rotor supported by two journal bearings with a magnetic control actuator. The results show the effectiveness of the vibration control, which is evaluated over the stable operating range of the controller.

The active magnetic bearing (AMB) is a relatively new technology which has many advantages compared with conventional bearing design. In an AMB system, the rolling-element back-up bearings are indispensable to protect the magnetic bearing rotor and stator, and other stationary seals along the rotor shaft. In this paper, a theoretical formulation is proposed and solved numerically to examine the transient response of the flexible rotor, from the time just previous to when the AMB shuts down and including the rotor drop onto the back-up bearing. The backward whirl of the rotor, which may lead to the destructive damage of the machinery, has been analytically predicted at very light support damping and very high support damping. Also, the vibration due to the nonlinearity of the contact point geometry has been included in the analysis. The influence of the support damping on the displacement of the disk and also the contact force between the journal and the inner-race of the back-up bearing have been computed for various rotor system parameters. By comparing these results with the optimum support damping for the simple flexible rotor model, it is shown that this support damping optimization can be applicable for specifying the required optimum range of support damping for the back-up bearings of AMB systems.

This study presents an investigation of the dynamics of a rotor system with bearing clearance. Of particular interest is the influence of such effects on coupled disk/shaft vibration. Experimental results for a rotor system with a flexible disk are presented and compared to predictions from a simulation model. Some insights and conclusions are obtained with regard to the conditions under which such vibration may be significant.

A direct numerical method for the determination of instability threshold and stability boundaries of flexible rotor-bearing systems is presented. The proposed procedure can also be used to improve the system stability by considering the design variables as operating parameters. The finite element method is utilized in the formulation of system equations of motion. The numerical algorithm is based on nonlinear optimization techniques. Two examples are presented to illustrate the feasibility, desirability, and ability of the proposed algorithm. A simple journal bearing system is used for the parametric study. An industrial high-speed compressor is employed to demonstrate the ability of this algorithm to deal with practical applications. The stability boundaries calculated from this algorithm are in agreement with the experimental results.

Parametric steady-state vibrations of an asymmetric rotor while passing through primary resonance and the associated stability behavior are analyzed. The undamped case is considered and the equations of motion are rewritten in a form suitable for applying the method of multiple scales. Sensitivity to the bearing as well as shaft asymmetries of the oscillations due to unbalance excitation is evaluated. Expressions for amplitude and frequency modulation functions are obtained and are specialized to yield the steady-state solutions near primary resonance. Frequency-amplitude relationships that result from combined parametric and mass unbalance excitations are derived. Stability regions in the parameter space are obtained based on the time evolution of the amplitude and phase of the steady-state motions. The effects of bearing asymmetry on the amplitude and phase of the resonant oscillations are brought out. The sensitivity of vibrational and stability characteristics to various rotor-dynamic system parameters is illustrated through a numerical investigation.

The recent growth of magnetic bearing applications in the chemical, utility and aerospace industries requires more accurate design tools to insure reliable and effective operation as components of the overall rotor bearing system. This paper provides a novel approach for simulating a flexible rotor suspended in magnetic bearings, accounting for eddy current effects. The manuscript discusses some magnetic bearing fundamentals, leading up to the new methods for modeling. Test results show very good correlation with theory and confirm the importance of considering eddy current effects. Surface conductivity and frequency dependent permeability effects on the B field in the magnet gaps are shown to be very significant on overall system stability. The results in the manuscript should be very useful to the design or test rotor dynamicist.

In a previous paper (Ehrich, 1994), the author has cataloged a variety of unique rotordynamic responses which have been observed in a computer model of a simple Jeffcott rotor in a nonlinear anisotropic mounting system; that is, operating eccentrically within a clearance and in local intermittent contact with the stator. In addition to the critical synchronous resonant response similar to that found in linear systems, unique responses are found at subcritical operating speeds - superharmonic pseudo-resonances, and regions of chaotic and periodic response in transition zones between successive superharmonic orders. In the transcritical operating regime on both the subcritical and supercritical sides of the critical peak, spontaneous sidebanding is found when the system is very lightly damped. At supercritical operating speeds, subharmonic pseudo-resonances, and regions of chaotic response and periodic response in transition zones between successive subharmonic orders are identified. These phenomena are characterized by their unique signature in the response curve, derived from a simple numerical model of a Jeffcott rotor with a bilinear stiffness in the direction normal to the plane of contact. Each phenomenon is further characterized by a typical example of the wave form and the wave form's spectral analysis. All the wave forms display the tendency of the nonlinear system to have a significant asynchronous response component at its natural frequency irrespective of the rotational or stimulus frequency. In a more recent publication (Ehrich, 1995), recorded observations of rotordynamic response, in the format of 'waterfall' or 'cascade' charts, of operational high-speed turbomachinery in two typical instances have been compared with equivalent data from the same computer model used to illustrate the cataloged phenomena. The instances are representative of several of the cataloged phenomena. Excellent correspondence between the computed waterfall charts and the data from the actual operational machinery are achieved. The results may be of considerable interest to the community of vibrations engineers who deal with prevention of and remedial action for deleterious asynchronous vibration in operational high speed rotating machinery.

The concept of using a statistically derived estimate of conductor motion size to determine stability was employed to obtain the stability properties of the rotor windings of a 70-MW superconducting generator in static conditions. Specifically, the frequency of quenching events during the excitation of rotor windings were approximated. In addition, good agreement between quench test results and numerical estimates of quenching behavior were obtained.

The standard lumped parameter model for flow in an eccentrically offset labyrinth seal, which assumes constant upstream and downstream boundary conditions, has been extended to include the effects of a nonuniform upstream cavity flow due to coupling. This new model predicts that the upstream perturbations in pressure and azimuthal velocity caused by this coupling can have a very strong impact on the pressure distribution in the seal gland itself. Augmentation by a factor of four, over the uniform inlet model, is predicted under some circumstances. Although no precise comparison to the experimental data with this new model was possible, due to the lack of control over the face seal venting the upstream cavity to the center hub plenum, the calculated effect of this coupling was shown to be approximately what was required to restore quantitative agreement between the data and theory. The new theory can explain the anomalously large pressure nonuniformity previously found by other authors in short seals as well as the first few glands of multicavity seals.

A theoretical and experimental investigation on the aerodynamic forces generated by a single gland labyrinth seal executing a spinning/whirling motion has been conducted. A lumped parameter model, which includes the kinetic energy carryover effect, is presented along with a linear perturbation solution technique. The resulting system is nondimensionalized and the physical significance of the reduced parameters is discussed. Closed-form algebraic formulas are given for some simple limiting cases. It is shown that the total cross force predicted by this model can be represented as the sum of an ideal component due to an inviscid flow with entry swirl and a viscous part due to the change in swirl created by friction inside the gland. The frequency-dependent ideal part is solely responsible for the rotordynamic direct damping. The facility designed and built to measure these frequency dependent forces is described. Experimental data confirm the validity and usefulness of this ideal/viscous decomposition. A method for calculating the damping coefficients based on the force decomposition using the static measurements only is presented.