Silicon based power micro-electro-mechanical system (MEMS) applications require high-speed microrotating machinery operating stably over a large range of operating conditions. The technical barriers to achieving stable high-speed operation with micro-gas-bearings are governed by (1) stringent fabrication tolerance requirements and manufacturing repeatability, (2) structural integrity of the silicon rotors, (3) rotordynamic coupling effects due to leakage flows, (4) bearing losses and power requirements, and (5) transcritical operation and whirl instability issues. To enable high-power density the micro-turbomachinery must be run at tip speeds comparable to conventional scale turbomachinery. The rotors of the micro-gas turbines are supported by hydrostatic gas journal and hydrostatic gas thrust bearings. Dictated by fabrication constraints the location of the gas journal bearings is at the outer periphery of the rotor. The high bearing surface speeds (target nearly 10 × 10 6 mm rpm ), the very low bearing aspect ratios ( L / D < 0.1 ) , and the laminar flow regime in the bearing gap ( Re < 500 ) place these micro-bearing designs into unexplored regimes in the parameter space. A gas-bearing supported micro-air turbine was developed with the objectives of demonstrating repeatable, stable high-speed gas-bearing operation and verifying the previously developed micro-gas-bearing analytical models. The paper synthesizes and integrates the established micro-gas-bearing theories and insight gained from extensive experimental work. The characteristics of the new micro-air turbine include a four-chamber journal bearing feed system to introduce stiffness anisotropy, labyrinth seals to avoid rotordynamic coupling effects of leakage flows, a reinforced thrust bearing structural design, a redesigned turbine rotor to increase power, a symmetric feed system to avoid flow and force nonuniformity, and a new rotor micro-fabrication methodology for reduced rotor imbalance. A large number of test devices were successfully manufactured demonstrating repeatable bearing geometry. More specifically, three sets of devices with different journal bearing clearances were produced to investigate the dynamic behavior as a function of bearing geometry. Experiments were conducted to characterize the “as-fabricated” bearing geometry, the damping ratio, and the natural frequencies. Repeatable high-speed bearing operation was demonstrated using isotropic and anisotropic bearing settings reaching whirl-ratios between 20 and 40. A rotor speed of 1.7 × 10 6 rpm (equivalent to 370 m/s blade tip speed or a bearing DN number of 7 × 10 6 mm rpm ) was achieved demonstrating the feasibility of MEMS-based micro-scale rotating machinery and validating key aspects of the micro-gas-bearing theory.

A 4.2-mm diameter silicon rotor has been operated in a controlled and sustained manner at rotational speeds greater than 1.3 million rpm and power levels approaching 5 W. The rotor, supported by hydrostatic journal and thrust gas bearings, is driven by an air turbine. This turbomachinery/bearing test device was fabricated from single-crystal silicon wafers using micro-fabrication etching and bonding techniques. We believe this device is the first micro-machine to operate at a circumferential tip speed of over 300 meters per second, comparable to conventional macroscale turbomachinery, and necessary for achieving high levels of power density in micro-turbomachinery and micro-electrostatic/ electromagnetic devices. To achieve this level of peripheral speed, micro-fabricated rotors require stable, low-friction bearings for support. Due to the small scale of these devices as well as fabrication constraints that limit the bearing aspect ratio, the design regime is well outside that of more conventional devices. This paper focuses on bearing design and test, and rotordynamic issues for high-speed high-power micro-fabricated devices.

A low-order model was created to analyze a small-scale gas bearing with a diameter of 4.1 mm, designed to spin at 2.4 million rpm. Due to microfabrication constraints, the bearing lies outside the standard operating space and stable operation is a challenge. The model is constructed by reference to Newton’s second law for the rotor and employs stiffness and damping coefficients predicted by other models. At any operating point it is able to predict (1) whether the journal can sustain stable operation, and (2) the whirling frequency of the journal. Analysis shows that the best way to operate the bearing is in a hybrid mode where the bearing relies on hydrostatics at low speeds and hydrodynamics at high speeds. However, in transitioning from hydrostatic to hydrodynamic operation, the model shows that the bearing is prone to instability problems and great care must be taken in scheduling the bearing pressurization system in the course of accelerating through low and intermediate rotational speeds.

An experimental and theoretical investigation has been conducted to evaluate the effects seen in axial-flow compressors when the centerline of the rotor is displaced from the centerline of the static structure of the engine. This creates circumferentially nonuniform rotor-tip clearances, unsteady flow, and potentially increased clearances if the rotating and stationary parts come in contact. The result not only adversely affects compressor stall margin, pressure rise capability, and efficiency, but also generates an unsteady, destabilizing, aerodynamic force, called the Thomas/Alford force, which contributes significantly to rotor whirl instabilities in turbomachinery. Determining both the direction and magnitude of this force in compressors, relative to those in turbines, is especially important for the design of mechanically stable turbomachinery components. Part I of this two-part paper addresses these issues experimentally and Part II presents analyses from relevant computational models. Our results clearly show that the Thomas/Alford force can promote significant backward rotor whirl over much of the operating range of modern compressors, although some regions of zero and forward whirl were found near the design point. This is the first time that definitive measurements, coupled with compelling analyses, have been reported in the literature to resolve the long-standing disparity in findings concerning the direction and magnitude of whirl-inducing forces important in the design of modern axial-flow compressors.

An experimental and theoretical investigation was conducted to evaluate the effects seen in axial-flow compressors when the centerline of the rotor becomes displaced from the centerline of the static structure of the engine, thus creating circumferentially nonuniform rotor-tip clearances. This displacement produces unsteady flow and creates a system of destabilizing forces, which contribute significantly to rotor whirl instability in turbomachinery. These forces were first identified by Thomas (1958. Bull. AIM, 71 , No. 11/12, pp. 1039–1063.) for turbines and by Alford (1965. J. Eng. Power, Oct., pp. 333–334) for jet engines. In Part I, the results from an experimental investigation of these phenomena were presented. In this Part II, three analytic models were used to predict both the magnitude and direction of the Thomas/Alford force in its normalized form, known as the β coefficient, and the unsteady effects for the compressors tested in Part I. In addition, the effects of a whirling shaft were simulated to evaluate differences between a rotor with static offset and an actual whirling eccentric rotor. The models were also used to assess the influence of the nonaxisymmetric static pressure distribution on the rotor spool, which was not measured in the experiment. The models evaluated were (1) the two-sector parallel compressor (2SPC) model, (2) the infinite-segment-parallel-compressor (ISPC) model, and (3) the two-coupled actuator disk (2CAD) model. The results of these analyses were found to be in agreement with the experimental data in both sign and trend. Thus, the validated models provide a general means to predict the aerodynamic destabilizing forces for axial flow compressors in turbine engines. These tools have the potential to improve the design of rotordynamically stable turbomachinery.

Several observations have been made in the Fourier spectra of high speed rotor-dynamic response of uniformly spaced frequency spikes on either side of key synchronous or subharmonic or superharmonic response frequencies. In instances where this so-called “sidebanding” could not readily be explained as the nonlinear interaction or combination tones of two distinct stimuli at slightly different frequencies, we have referred to this class of phenomena as spontaneous sidebanding . It is invariably noted that the sideband spacing frequency appears to be a whole number fraction (1/J) of the operating speed which suggests that the wave form is periodic and completes a full cycle every J rotations of the rotor. Using a numerical model of a rotor which simulates local contact with a stator in close proximity as a bilinear spring, several studies have been carried out to explore the circumstances for this spontaneous sidebanding. Two general classes of this type of response have been found in systems that are effectively single-degree-of-freedom: (A) For highly nonlinear systems, the chaotic-like response in transition zones between successive orders of subharmonic and superharmonic operation is actually periodic, with a repetition index (J), and results in spontaneous sidebands clustered around the key subharmonic or superharmonic frequencies. No systematic relationship has been determined for the value of (J). (B) In transcritical operation of highly nonlinear and very lightly damped systems, a major sideband frequency spike is noted at a frequency which is approximately the system’s natural frequency. Recognition of this fact permits a simple estimate of the repetition index (J). All these observations from operation of the numerical model have been compared with experimental data derived from incidents of spontaneous sidebanding on aircraft gas turbine rotors. Excellent qualitative agreement has been found in most instances.

Subharmonic response in rotordynamics may be encountered when a rotor is operated with its rotational centerline eccentric to that of a close clearance static part, so that local contact can take place during each orbit when the rotor is excited by residual unbalance. The rotor will tend to bounce at or near its fundamental frequency when the rotor is operated at or near a speed which is a whole number [n] times that frequency. Using a simple numerical model of a Jeffcott rotor mounted on a nonlinear spring, it is found that the vibratory response in the transition zone midway between adjacent zones of subharmonic response has all the characteristics of chaotic behavior. The transition from subharmonic to chaotic response has a complex substructure which involves a sequence of bifurcations of the orbit with variations in speed. This class of rotordynamic behavior was confirmed and illustrated by experimental observations of the vibratory response of a high-speed turbomachine, operating at a speed between 8 and 9 times its fundamental rotor frequency when in local contact across a clearance in the support system. A narrow region between zones of 8th order and 9th order subharmonic response was identified where the response had all the characteristics of the chaotic motion identified in the numerical model.

In industries like the aircraft gas turbine, trans-critical or super-critical operating speeds are quite common, and rotating machinery must be mass produced with very demanding precision in balance effectiveness, generally without recourse to high-speed balance procedures. A procedure has been developed which permits high-speed multi-plane (i.e., three or more plane) balance correction to be made on rotors in simple conventional low-speed balance machines (Patent Applied For). The procedure accomplishes most of the benefits of actual high-speed or modal or true multi-plane balancing by utilizing other known or available data on the particular rotor’s generic dynamic behavior (i.e., its natural or critical mode shapes) and data on the particular rotor’s generic design and manufacture (i.e., its perceived generic patterns of unbalance distribution). For ( N ) balance planes, the procedure involves the specification of a Balancing Rule wherein a sequence of ( J ) low speed balance steps is specified (where J equals the integer part of [( N + 7)/2]). At each of these steps, some fraction (called a Balancing Factor) of the measured two plane unbalance vectors is applied to one or two of the other balance correction planes, before final correction is made on the last two correction planes themselves. A procedure is derived to predetermine those ( I ) Balance Factors (where I equals [ N -2]). The procedure involves an iterative sequence for computing the optimized Balance Factors, with convergence driven by the Newton Raphson procedure, and requires the specification of ( I ) pairs of generic unbalance distributions and natural mode shapes. The analytically derived Balancing Factors are designed to null the vibration response of the rotor excited by each of the specified generic unbalance distributions at the critical speed associated with the specified mode shape with which the generic unbalance distribution is paired.

Subharmonic vibration refers to the response of a dynamic system to excitation at a whole-number multiple (n) of its natural frequency by vibrating asynchronously at its natural frequency, that is, at (1/n) of the excitation. The phenomenon is generally associated with asymmetry in the stiffness vs. deflection characteristic of the system. It may be characterized as the “bouncing” of the rotor on the surface of the stiff support, energized by every nth unbalance impulse prior to contact. Second, third and fourth order subharmonic vibration responses have previously been observed in high speed rotating machinery with such an asymmetry in the bearing supports. An incident is reported where 8th and 9th order subharmonic vibration responses have been observed in a high speed rotor. A simple but exact computer model of the phenomenon has been evolved based on the numerical integration of a finite difference formulation. Response curves and wave forms of rotor deflection at individual speeds are computed. It is shown that the response is a series of pseudo-critical peaks at whole-number multiples of the rotational speed. Very high orders of subharmonic vibration are found to be possible for systems with low damping and extreme nonlinearity.

A vibration incident on a gas turbine engine was noted where two major excitation frequencies were involved—an excitation synchronous with rotor rotation, associated with rotor unbalance, and an asynchronous excitation associated with fluid inadvertently trapped in the rotor. Spectral analysis of the vibration wave form revealed not only the two base excitation frequencies, but also a component at the difference frequency. A mechanism for generating such a difference frequency is hypothesized—the truncation of the basic “beat frequency” wave form by virtue of clearance in the rotor bearing system. Fourier analysis of the hypothesized excitation wave form indicates that components at difference frequency are indeed generated, and also at the sum frequency and a spectrum of higher harmonics and side band frequencies. The hypothesized wave form’s spectral analysis bears a remarkable resemblance to the measured spectrum, except that low frequencies appear to have been greatly amplified in the experimental case, and high frequencies attenuated. This latter fact is attributed to the transmission characteristics of the gas turbine stator system, and is probably responsible for the lack of precise correspondence between the measured and hypothesized wave forms.

A systematic analysis is made of the potential instability of branched diffuser systems such as are inherent to the annular combustor systems of gas turbine engines. The system is modeled to include diffusion in each branch with pressure recovery characteristics which are a simple function of the fraction of total flow into the branch. Also included in each branch is a volume (i.e., an accumulator) and a resistance representing the combustor shells before the two branched streams rejoin. Analysis of the system equations is carried out in two perspectives. First, the equations are linearized and simple generalized criteria for stability are derived. Then the full nonlinear equations are programmed for digital computation in a form where they can be integrated with respect to time, and the full dynamic behavior of flows and pressure described. The computation is carried out several times for system variations of shell resistance, accumulator volumes, and diffuser characteristics, so that general conclusions may be drawn of the effects of these parameters on the frequency, wave form, and amplitudes of the systems’ oscillations.

The classic phenomenon of “dry friction whip,” generally associated with unlubricated journal bearings, is here reconsidered as playing an important role in the dynamic stability, and consequent integrity of radial rubs in all close clearance rotating machinery, particularly in turbomachinery elements such as labyrinth seals and blade tips. A simple analysis is completed for large (runaway) amplitudes of whipping on an analytic model which includes the stator as an independent dynamic system. Whirl frequencies are computed as a function of rotor and stator natural frequencies and damping. A stability criterion is developed as a function of these same variables. Testing on a simple experimental model gives general qualitative agreement to predicted trends, but is not conclusive quantitatively, probably because of the difficulty in simulating pure Coulomb friction at the rubbing interface. The simple generality that can be inferred from the set of derived stability criteria is that the broadest band of whip-free rubbing is achieved if rotor and stator dampings are made close to one another, and if the rotor and stator natural frequencies are kept dissimilar. Systems with identical rotor and stator natural frequencies are always unstable, and will whip at that same natural frequency. Systems with large stator damping will whip at rotor natural frequency. Systems with large rotor damping will whip at stator natural frequency.

Applying the Campbell [4] criterion (for excitation of axisymmetric physical systems in proximity to turbomachinery) to an analytic model of an annular acoustic cavity, a detailed view of acoustic resonances in the cavity is derived. The perspective expands on the prior work of Tyler and Sofrin [1] and others by considering the entire spectrum of modes, and the effects of axial Mach number and swirl velocity in the inlet cavity, as well as the geometric proportions of the cavity. Noise data from an experimental turbomachinery vehicle are examined over a spectrum of rotative speeds by means of a narrow band filter tracking integer multiples of rotative speed. Many aspects of the predicted acoustic resonances at supersonic wheel speeds are substantially confirmed. Additionally, unexpected subsonic resonant peaks are observed and partially explained. Conclusions are drawn as to probable influence of the aerodynamic and geometric parameters in manipulating acoustic resonances out of the operating range of aircraft engines.

Fluids are often introduced accidentally and trapped inside of high speed rotors. This has given concern as to the possible deleterious effects of such trapped fluids on the system’s vibration characteristics. Trapped fluids will induce asynchronous vibratory motion of high speed rotors at supercritical rotational speeds. This tendency is examined in analytic detail. Assuming that the circumferential velocity of the trapped fluid varies linearly with radius, the generalized shape of the fluid film is derived. Integrating the fluid pressure on the cavity walls gives the net fluid forces and permits computation of whirl frequency and whirl amplitude as functions of rotative speed. Generalized plots are given of the film geometry, of the rotative speed at which asynchronous whirl starts, and of the induced whirl speed. General response curves are also given, showing whirl amplitude as a function of rotative speed. The detailed results indicate that whirl occurs at a rotative speed approximately double the induced whirl frequency (as happens with many rotor whirl mechanisms). Higher values of the system parameter g result in somewhat lower whirl onset speeds. The whirl velocity is approximately equal to the rotor critical speed, or slightly lower for large masses of trapped fluid. Rotor whirl amplitude increases sharply with rotative speed above onset speed until a limiting condition where the fluid film (and analytic solution) break down, at a rotative speed about 6 percent above onset speed. Trapped fluids will also influence the normal synchronous vibrations induced by rotor unbalance. Analyses of a simple model of synchronous, solid body rotation are made which give exact solutions for the condition of a fully welted cavity periphery, and give approximate solutions for a partially welted periphery. It is concluded that the trapped fluid generally reduces the critical frequency, reduces critical amplitude, and reduces high speed (supercritical) amplitude. The effect is quite small with a partially welted circumference, but is surprisingly large in the case where the periphery of the cavity stays fully welted at maximum vibration amplitude. In this case, the rotor acts as if the cavity were entirely full of fluid, even though actual trapped fluid may fill only a small fraction of the cavity volume! Significant reductions in critical frequencies and amplitude are thereby observed. In the case of systems that are partially welted in going through their resonance peak, a “rightward leaning” resonance peak is observed which results in jump phenomena and hysteresis in amplitude on accelerating and decelerating through the peak.