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This paper presents a theoretical model for analytical study of internal friction-induced whirl instability due to shrink fit and interference fit joints in rotating machines. It is the theoretical continuation of an experimental study on the effects of internal friction published in a companion paper. The theoretical study shows that the follower force internal friction model proposed by Gunter in 1965 is a physically incorrect model due to its inconsistency with Newton’s Third Law of Motion. The internal moments model proposed by Lund in 1976 to explain unstable sub-synchronous vibrations due to micro-slip in interference and shrink fit joints is physically consistent with the basic principles of mechanics and can explain rotordynamic instability due to internal friction in rotating machines. Rotordynamic simulations of the experimental single-disk and two-disk rotor based on an internal moments model are carried out using finite element based rotordynamics software. Rotordynamic simulations of the unstable experimental rotors using the linear viscous form of the internal moments model, with selected moment stiffness and damping parameters at the interface, result in correct prediction of the instabilities (negative real part of the damped eigenvalues).

This paper presents an experimental study of internal friction effects on two different test rotors with shrink fit and interference fit joints. One of the setups was a single-disk rotor with an interference fit joint to the main shaft through a mechanically adjustable tapered sleeve; the other setup was a two-disk rotor with a variable cross-section shaft, having the two disks shrunk fit into an aluminum sleeve at the ends. The slippage in these types of joints is known to be a potential source of sub-synchronous rotordynamic instability in rotating machines. The experimental results from the rotor setups show the presence of strong unstable sub-synchronous rotor vibrations initiated at operating speeds above the first critical speeds of the rotor-bearing systems. The experiments with the two-disk rotor demonstrated that different distributions of shrink fit at two locations (loose fit at one end; tight fit at the other end) were necessary to produce rotordynamic instability. The instability in that case caused complete wreckage of the test rig, showing that the sub-synchronous vibrations caused by slippage at interference fit joints are potentially dangerous to the safe operation of rotating machines. Furthermore, a critical radial interference (shrink fit) of 1 mils (25.4 μm) at speed was identified to cause an onset of instability in several of the experiments.Copyright © 2008 by ASME

Rotordynamic instability, commonly observed as subsynchronous vibration, is a serious problem that can cause heavy damage to a turbomachine or make it incapable of operation due to high vibration levels. However, all subsynchronous vibrations are not necessarily unstable. A way to quickly diagnose them would be helpful. In an earlier paper, the authors presented data from experiments that simulated various causes of sub-synchronous vibrations, some causes being genuine rotordynamic instabilities and some others being benign (stable), and identified ways to diagnose and classify the subsynchronous motions. In a continuation of the same study, subsynchronous vibrations due to coupled lateral-torsional effects are experimentally simulated, the objective being to signal-analyze these vibrations to find unique signatures that identify this cause and also be able to recognize if they are a true rotordynamic instability or not. To this end, a test rig was built with parallel shafts coupled by gears, driven by a DC motor at one end and loaded at the other end, to closely simulate a real-world machine. A torsional mathematical model for the test rig is also presented to predict its torsional natural frequencies. Experiments were conducted wherein the first torsional natural frequency was externally excited, with the shaft spinning at a higher speed. The result was a false sub-synchronous “instability” signal in the lateral measurements. A method to distinguish these vibrations from a genuine lateral non-synchronous instability is presented. Also, a new diagnostic method to classify the subsynchronous vibration as benign is elucidated.Copyright © 2008 by ASME

Rotordynamic instability can be disastrous for the operation of high speed turbomachines in the industry. Most ‘instabilities’ are due to de-stabilizing cross coupled forces from variable fluid dynamic pressure around a rotor component, acting in the direction of forward whirl and causing subsynchronous orbiting of the rotor. However, all subsynchronous whirling are not unstable and methods to diagnose the potentially unstable kind from the benign are critical to the health of the rotor-bearing system. In this study, methods to demarcate between the two are detailed. Orbit shape, “frequency tracking” and agreement of subsynchronous frequencies with known eigenvalues are used as diagnostic tools. It is shown that a change in synchronous phase angle produced by de-stabilizing cross coupled forces can be used as a definitive indicator of incipient instability. Typical signatures of subharmonic vibrations induced from non-linear stiffness of the rotor- bearing system are examined analytically and through experiments.Copyright © 2007 by ASME

Rotordynamic instability, commonly observed as subsynchronous vibration, is a serious problem that can cause heavy damage to a turbomachine or make it incapable of operation due to high vibration levels. However, all subsynchronous vibrations are not necessarily unstable. If the amplitude of the subsynchronous vibration is large, it can cause damage to seals, bearings, or process wheels. If it is small, the question arises as to whether it has the potential to grow larger (“instability”) or whether it is benign and harmless. A way to know would be helpful. The objective of this study is to signal-analyze subsynchronous vibration in turbomachinery and distinguish benign subsynchronous vibration from true rotordynamic instability. Effort is also made to identify unique signatures to a cause, thereby aiding in faster diagnosis. A computer simulation study is conducted on four rotors, including two gas-reinjection compressors that went unstable, to examine the possibility of using the change in synchronous phase angle as a possible indicator of impending instability. Two other rotors in laboratory test rigs were also studied, both experimentally and with computer simulations of the phase angle response. The computer simulations and experimental results agree very closely on the test rigs. Measured signals from another rotor, with bearings having a dead-band clearance are studied as an example of a benign cause that can result in subsynchronous vibration. The effect is studied with the rotor in both horizontal and vertical positions and clear indicators are discovered that confirm the subsynchronous vibration to be benign in nature. Signatures from another rotor with internal friction are also presented. The experimental data clearly shows indicators that distinguish it as a genuine instability. Dry friction whip is also experimentally produced on a test rig. The measurements show that dry friction whip defies all general rules of thumb for diagnosing a true instability and at the same time is a very violent one as well.Copyright © 2007 by ASME

The effects of two seal design parameters, namely blade (tooth) thickness and blade profile, on labyrinth seal leakage, as well as the effect of operating a seal in an off-center position, were examined through a series of non-rotating tests. Two reconfigurable seal designs were used, which enabled testing of two- four-, and six-bladed see-through labyrinth seals with different geometries using the same sets of seal blades. Leakage and cavity pressure measurements were made on each of twenty-three seal configurations with a four inch (101.6 mm) diameter journal. Tests were carried out with air as the working fluid at supply pressures of up to 100 psi-a (6.89 bar-a). Experimental results showed that doubling the thickness of the labyrinth blades significantly influenced leakage, reducing the flow-rate through the seals by up to 20%. Tests to determine the effect of blade-tip profile produced more equivocal results, with the results of experiments using each of the two test seal designs contradicting each other. Tests on one set of hardware indicated that beveling blades on the downstream side was most effective in limiting leakage whereas tests on newer hardware with tighter clearances indicated that seals with flat-tipped blades were superior. The test results illustrated that both blade profile and blade thickness could be manipulated so as to reduce seal leakage. However, an examination of the effects of both factors together indicated that the influence of one of these parameters can, to some extent, negate the influence of the other (especially in cases with tighter clearances). Lastly, for all configurations tested, results showed that leakage through a seal increases with increased eccentricity and that this phenomenon was considerably more pronounced at lower supply pressures.Copyright © 2007 by ASME

One of the main contributors to rotordynamic instability in modern day turbomachinery is cross-coupled stiffness. The terminology comes from the phenomenon in which the reactive stiffness force is normal to the rotor displacement. Cross-coupled stiffness (CCS) can be generated from fluid pressure forces in internal machine components such as gas/liquid seals, hydrodynamic bearings, axial flow turbomachinery stages, and centrifugal impellers. Some of these machine components exhibit a type of CCS that materializes as a follower-force, which is normal to the whirl orbit radius and therefore tangent to a circular whirl orbit. With the follower force collinear with the whirl velocity, energy is fed into the motion and makes it grow larger with time. This destabilizing force can be represented in a Newtonian coordinate system by opposite-sign CCS, Kxy = −Kyx. However, there are cases of CCS where the forces produced are not always in the direction of rotor whirl, and the net work of the force per cycle is zero. These CCS forces are purely distortive to the orbit and are not de-stabilizing. They are represented by same-sign CCS, Kxy, Kyx, not necessarily equal. This paper investigates the effects of same-sign CCS on rotordynamics. Experiments are presented that show the measurement of same-sign CCS on a pocket damper seal and the required testing scheme to determine the effect on rotordynamic stability.

In a previous ASME paper the second author reported experiments on wire mesh bearing dampers (WMD) incorporated in a power turbine rotor-bearing system in order to enable a direct comparison between WMD and squeeze film dampers (SFD). The results showed that both WMD and SFD perform equally well for reducing the rotordynamic amplitudes of vibration. Moreover the WMD were found to have significant advantages over SFD. The damping provided by the wire mesh is independent of temperature changes and presence of turbine oil. Experiments by another investigator showed that WMD are capable of sustaining more than twice the unbalance as compared to SFD, which promises possible application to withstand blade loss loads. This paper presents empirically developed non-dimensional design equations for WMD, capable of predicting stiffness and damping for a wire mesh ‘donut’ subject to changes in various design, installation, and operational parameters.Copyright © 2005 by ASME

A model of impact of a tennis ball with a flat surface is developed based on a planar, two-mass, linear, four degree of freedom vibration system idealization. The impact is assumed to be incident on a flat surface with friction. The incident parameters of the ball include the centre of mass translational velocity, angle of impact with the surface and the incident angular spin of the ball. The linear, piecemeal vibration model predicts the corresponding rebound parameters of the tennis ball. The model also predicts the duration of contact of the tennis ball with the flat surface, the transition of motion of the tennis ball during contact with the ground from sliding to rolling contact, and the resulting contact forces developed between the tennis ball and the flat surface. The model is computationally efficient because the governing differential equations of motion are linear and their standard solutions can be easily implemented on a personal computer. Predictions of the rebound parameters from the model are compared with experimental findings on tennis balls which are incident on a flat surface with various angles, velocities and angular spins (zero spin, topspin and backspin). For selected parameters of the two-mass model, the comparisons show excellent agreement between the model and the measurements.

Pocket damper seals (PDSs) have been shown to offer an improved rotordynamic performance compared to conventional labyrinth seals, providing large amounts of direct damping and small cross-coupled coefficients. To this date, the rotordynamic characteristics of the PDSs have been determined using linearized models, with validity limited to small displacements of the rotor. The present investigation determines the effect of the nonlinear forces arising from the pocket damper seal on the response and stability of a PDS-rotor system when subjected to large dynamic excitations. It is determined that the nonlinear forces maintain bounded amplitudes of vibration (“limit cycles”) after the rotor passes the threshold speed of instability predicted by the linear theory. It is also found that the critical speed of the PDS-rotor system is well predicted by using both linear and nonlinear models of the seal forces, as both predictions compare well with the experimental measurements.Copyright © 2003 by ASME

Pocket damper seals can produce large amounts of damping in turbomachines where the working fluid is compressible. Even with non-optimum design configurations, they have solved rotordynamic instability problems in high-pressure compressors. They do not rely on viscous effects, so they are ideal for oil-free applications. It is shown here that proper sizing of the seal pockets can maximize the damping. Significant parameters and variables have been identified, both experimentally and theoretically, that reflect the fundamental effects of the pocket damper seal to produce rotordynamic damping. Dimensionless analysis was performed, resulting in significant Pi groups involving damping, stiffness and leakage parameters. Of particular interest is a Helmholtz Pi group that shows 1) the benefit from optimizing the volume of the pockets around the seal, and 2) the frequency dependence of the seal. Dimensionless prediction equations are shown graphically. The graphs were generated by a computer code that uses a single control volume model. The code has been verified experimentally on several different seal test rigs with data taken at different vibration frequencies spaced an octave apart.Copyright © 2002 by ASME