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FOREWORD

J. Tribol. 1987;109(1):1. doi:10.1115/1.3261320.
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Abstract
Commentary by Dr. Valentin Fuster

RESEARCH PAPERS

J. Tribol. 1987;109(1):2-15. doi:10.1115/1.3261322.
Abstract
Topics: Lubrication
Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):21-36. doi:10.1115/1.3261321.
Abstract
Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):37-41. doi:10.1115/1.3261324.

The development of the concept of spring and damping coefficients for journal bearings is briefly reviewed. Methods for computing the coefficients are described, and their use in rotor dynamics calculations (unbalance response, stability) is discussed. The limitations imposed by nonlinearities on the application of the coefficients is illustrated by examples.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):42-49. doi:10.1115/1.3261325.

The governing equation of lubrication theory was developed by Osborne Reynolds in 1886. Two-dimensional solutions to this equation were not obtained until many years afterwards, first by electric analog techniques and then by digital computer. Solving the equation has produced tremendous advances in bearing design and development and insight into the mechanisms of lubricating film behavior. This paper concentrates on some of the developments for solving the Reynolds’ equation and some applications that have followed a better understanding of fluid-film phenomena.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):50-53. doi:10.1115/1.3261326.
Abstract
Topics: Reflection
Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):54-59. doi:10.1115/1.3261327.

A modified form of Reynolds’ equation is obtained by integrating the thin film equations across the film and approximating the nonlinear inertia terms using the lubrication theory velocity field. The resulting nonhomogeneous partial differential equation is tackled by the classical separation of variables technique, and the corresponding eigenvalue problem solved by a computer library routine. The end boundary condition is that the local exiting flow rate is proportional to the pressure difference across the end seal. The results show a large effect of inertia in terms of pressure amplitude and phase shift, but the phase shift is reduced at large eccentricity. End leakage and bearing length change the pressure amplitude for the inertialess and inertia cases, but in equal measure. Hence conclusions reported in the literature regarding the effect of inertia in long or short dampers generally hold for finite bearings.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):60-64. doi:10.1115/1.3261328.

The influence of fluid inertia on a plane slider bearing that operates at a 0(1) modified Reynolds number is examined in this study. The flow is laminar, and the Reynolds number—based on the slider velocity, lubricant kinematic viscosity, and leading-edge slider height—can be as high as 1000. Our major conclusion is that the primary effect of fluid inertia is to raise the pressure boundary condition near the bearing leading-edge. Lubrication theory is used to determine the pressure in the fluid film in the region downstream of the bearing entry. The leading-edge pressure increase caused by convective inertia is determined by a mass-flux balance between the flow near the leading-edge, and the flow through the bearing gap, which is determined by lubrication theory. Analytical results are obtained both for the convective-inertia pressure at the bearing entrance and for the pressure under the slider bearing. Results are compared to other numerical calculations and to analytical results, where the fluid inertia terms were kept throughout the bearing gap.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):65-70. doi:10.1115/1.3261329.

An approximate method is developed to include the flexibility of the pad in the calculation of the stiffness and damping properties of a tilting pad journal bearing. It is a small-amplitude perturbation solution in which the pad deformation is accounted for solely by the change in clearance. A comparison of results with those obtained from a more complete elasto-hydrodynamic solution shows good agreement.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):71-76. doi:10.1115/1.3261330.

The performance of a cylindrical bore bearing fed by two axial grooves orthogonal to the load line is analyzed by solving the Navier-Stokes equations using the finite element method. This produces detailed information about the three-dimensional velocity and pressure field within the hydrodynamic film. It is also shown that the method may be applied to long bearing geometries where recirculatory flows occur and in which the governing equations are elliptic. As expected the analysis confirms that lubricant inertia does not affect bearing performance significantly.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):77-82. doi:10.1115/1.3261331.

Several cases of cylindrical bearings lubricated with ferrofluid are examined by means of a finite-difference numerical procedure. The subregion of cavitation is determined by imposing mass conservation across the “rupture” and “reformation” boundaries of the complete film. The examined cases refer to bearings confined by sealing rings formed by the lubricant itself, for which there is no need of external supply. In each case the most significant parameters are calculated and it is possible to see that the use of these bearings is favorable when speeds are low, clearances large, and loads light, because in such conditions the magnetic effects are comparable with the pure hydrodynamic ones and the load capacity results appreciably higher.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):83-85. doi:10.1115/1.3261332.

It is shown that the difference between the forces acting on the top and bottom plates of a squeeze-film bearing, or the transmission loss, is entirely an inertia phenomenon and arises due to the acceleration of the top plate (the bottom plate is stationary). Its magnitude is given by half of the mass of the fluid film multiplied by the top plate acceleration. This is valid for Newtonian and some Oldroyd-B fluid provided that edge effects are negligible and no bifurcation of solution occurs.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):86-90. doi:10.1115/1.3261333.

The modified Reynolds equation in conjunction with the modified Coyne-Elrod rupture model is used to investigate the inertia effect on the pressure distribution in converging-diverging bearings. The modified Reynolds equation is solved analytically for infinitely long bearings, including the cylinder-plane bearing and the journal bearing. The results showed that the fluid inertia tends to stretch the fluid film and to move the film rupture point farther downstream. The effects are profound even at a moderate value of the reduced Reynolds number, Re* ≈ 0.13 based on the minimum film thickness.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):91-99. doi:10.1115/1.3261334.

A general theory is presented for the calculation of leakage rates and quasi-static opening force for aligned face seals operating in a turbulent, two-phase flow regime. Here, the fluid is assumed to be a homogeneous mixture of liquid and vapor in thermodynamic equilibrium. Full account of inertial effects and heat generation due to viscous dissipation is made. Solution of the model governing differential equations is accomplished through the use of a fourth-order Runge-Kutta numerical integration scheme. Several numerical examples are presented. The calculation of mass leakage rates and opening force under choked and unchoked conditions are discussed and the phenomenon of all-liquid choked flow is explained. Opening force curves are generated as a function of gap height for some situations of interest and their relation to seal axial stiffness is examined.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):100-107. doi:10.1115/1.3261298.

The flow of a homogeneous incompressible non-Newtonian fluid of the differential type between infinite eccentric rotating cylinders is discussed within the context of the lubrication approximation. The problem is studied by means of a perturbation and the effects of the non-Newtonian parameters are delineated. It is found that the load carrying capacity of the bearing can be significantly altered by the non-Newtonian character of the fluid.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):108-114. doi:10.1115/1.3261299.

This paper offers an analysis and computer solutions for misaligned thrust faces, including thermal effects and proper cavitation boundary conditions. The geometries analyzed range from full 2π arcs to pads of 27 deg angular extent, for both parallel surfaces and standard tapered-land thrust bearings. It is shown that, except for parallel surfaces, no cavitation occurs in thrust bearings, even under severe misalignment; that the gain in hydrodynamic force versus a decrease in minimum film thickness is most favorable at low and moderate levels of misalignment; and that for high load capacity, small arcs (many pads) are preferred to large ones.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):115-123. doi:10.1115/1.3261302.

A new direct iterative method for obtaining the time-varying behavior of a statically indeterminate shafting system within one of its hydrodynamic journal bearings is described. A modified Newmark’s method is used to step in time. At each integration time step an optimization technique iterates between the shafting system and the oil film analyses until an equilibrium is achieved. The three-dimensional shafting system structural analysis and the two-dimensional oil film hydrodynamic analysis utilize the finite element method. The “hourglass control” method is employed for the construction of the oil film fluidity matrix. A numerical example illustrates the method.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):124-128. doi:10.1115/1.3261303.

Test results consisting of direct and transverse force coefficients are presented for eleven, sawtooth-pattern, damper-seal configurations. The designation “damper” seal refers to a seal which uses a deliberately roughened stator and smooth rotor, as suggested by von Pragenau [1], to increase the net seal damping force. The designation “sawtooth-pattern” refers to a stator-roughness pattern whose cross section normal to the axis of the seal resembles saw teeth with the teeth direction opposing fluid motion in the direction of shaft rotation. The sawtooth pattern yields axial grooves in the stator which are interrupted by spacer elements which act as flow constrictions or “dams.” Sawtooth-pattern seals had more damping than smooth seals but less than the round-hole-pattern seals tested previously. Stiffness of sawtooth and round-hole-pattern seals were comparable. Leakage of maximum-damping configurations was greater for sawtooth-pattern than for round-hole-pattern seals; both types of seals leaked substantially less than did smooth seals. If damping is sacrificed, sawtooth-pattern seals can be designed to leak less than round-hole-pattern seals.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):129-135. doi:10.1115/1.3261304.

The linearized fluid film dynamic coefficients, i.e., stiffness and damping, of flexibly-mounted rotor noncontacting mechanical face seals are found. The coefficients are derived from a previous study where the flexibly mounted element was the stator. The two cases of inward and outward flows, both having converging gaps in the direction of flow, are analyzed for the two mounting configurations, and it is found that the later case possesses higher angular stiffness.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):136-143. doi:10.1115/1.3261305.

An analysis for helically-grooved turbulent annular seals is developed to predict leakage and dynamic coefficients, as related to rotordynamics. The grooved surface pattern is formulated as an inhomogeneous directivity in surface shear stress. The governing equations, based on both Hirs’ turbulent lubrication theory and “fine-groove” theory, are expanded in the eccentricity ratio to yield zeroth and first-order perturbation solutions. The zeroth-order equations define the steady-state leakage and the circumferential velocity development due to wall shear for a centered rotor position. The first-order equations define perturbations in the pressure and axial and circumferential velocity fields due to small motion of the rotor about the centered position. Numerical results are presented for proposed grooved seals in the High Pressure Oxygen Turbopump (HPOTP) of the Space Shuttle Main Engine (SSME) and for a water-pump application. The results show that an optimum helix angle exists from a rotordynamic stability viewpoint. Further, a properly designed helically-grooved stator is predicted to have pronounced stability advantages over other currently used seals.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):144-148. doi:10.1115/1.3261306.

The rotordynamic coefficients of an incompressible-flow annular pressure seal were determined using a bulk-flow model in conjunction with two different friction factor relationships. The first, Hirs’ equation, assumes the friction factor is a function of Reynolds number only. The second, Moody’s equation, approximates Moody’s diagram and assumes the friction factor is a function of both Reynolds number and relative roughness. For each value of relative roughness, Hirs’ constants were determined so that both equations gave the same magnitude and slope of the friction factor. For smooth seals, both relationships give the same results. For rough seals (e/2H0 = 0.05) Moody’s equation predicts 44 percent greater direct stiffness, 35 percent greater cross-coupled stiffness, 19 percent smaller cross-coupled damping, 59 percent smaller cross-coupled inertia, and nominally the same direct damping and direct inertia.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):149-154. doi:10.1115/1.3261307.

Squeeze film dampers are frequently used for stabilization and/or vibration control of rotating machinery. Theoretical analyses to date generally assume an incompressible lubricant. In practice, however, depending on the capacity of the lubricant reservoir, the lubricant at damper inlet contains varying amounts of dissolved gas, which come out of solution to form a “spongy” gas-liquid mixture during damper operation. This paper examines theoretically and experimentally the effects such entrained gases have on damper performance, particularly on damper load capacity and the likelihood of multistable operation. It is shown that under certain operating conditions, a significant delay in the onset of bistable operation is predicted, depending on the fluid film model employed. Preliminary tests indicate that at low bearing parameter values (B ≐ 0.02), the homogeneous compressible film model using the Hayward rather than the Isbin viscosity relationship for gas-liquid mixtures provides the best prediction of damper performance. Of the incompressible film models, the zero pressure truncation predictions are generally quite satisfactory and superior to the commonly used π-film predictions.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):155-163. doi:10.1115/1.3261308.

A technique for obtaining estimates of the direct damping and inertial coefficients of a squeeze-film bearing is described. This involves applying parametric identification and optimization techniques to digitized, free-decay experimental displacement records. The experimentally obtained coefficients, derived by this technique, were found to be significantly higher in magnitude (in some cases by a factor of about 10) than the corresponding values derived from conventional short-bearing theory, and to be virtually independent of the frequency of vibration.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):164-168. doi:10.1115/1.3261311.

The one-dimensional squeeze film damper is modeled for high speed flow by using the two-equation (k-ε) turbulent transport model. The assumption is made that the fluid flow at each local region of the squeeze film damper has similar behavior to inertialess flow in a channel at comparable Reynolds number. Using the k-ε model, the inertialess channel flow case is solved. Based on this result, correlations are obtained for the mean velocity, inertia and viscous terms of the integrated momentum equation for the squeeze film damper. It is found that turbulence increases the magnitude of the fluid pressure and the tangential force, while fluid inertia causes a shift on the pressure creating a significant radial force. In applications, turbulence may be a beneficial effect, increasing the principal damping force; while inertia may be detrimental increasing the cross-coupling forces.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):169-176. doi:10.1115/1.3261312.

The nonsynchronous motion of a rigid rotor-squeeze film damper system was investigated. This research classifies the phenomenon of the system with perfectly preloaded centering springs, reveals its existence conditions, and analyzes the system’s topological structures associated with the phenomenon. When the centering springs of the system do not perfectly balance the gravity load, a unidirectional gravitational residual is resulted. This research also studied the effect of the gravitational residual on the system’s behavior and the degree of stability for the various topological structures of the corresponding autonomous system. It shows that the residual improves the performance of the system because it either suppresses the nonsynchronous orbit of the system or changes it into various subharmonic orbits. This research shows that though the squeeze film damper system is inherently stable, it can have some undesirable nonsynchronous behavior for a wide range of system parameters.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):177-182. doi:10.1115/1.3261314.

The working characteristics of a finite width slider bearing lubricated by a non-Newtonian fluid are computed. The analysis proposed here allows its performances to be evaluated by means of a pocket calculator. For that purpose, a computer code based on a finite element method is used. The program runs for different values of pertinent kinematical, geometrical and rheological parameters. The corresponding results are fitted by means of adequate analytical formulas, which are very easy to handle. The accuracy of these empirical formulas is investigated in several typical cases. The agreement with the numerical solution is proven to be satisfactory.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):183-188. doi:10.1115/1.3261316.

The dynamic stability of a blocked center inward pumping spiral grooved thrust bearing is investigated. For this purpose two methods are considered comparatively, namely a standard small perturbation one, and an extension of the method used previously to determine air-hammer phenomena in externally pressurized gas bearings. The first method gives a more detailed description of situations in which the film is stable or unstable, while the second one gives a limit for the speed (or compressibility parameter Λc ) up to which the bearing is unconditionally stable. The second method is simpler and of practical interest since at higher speeds the critical mass furnished by the first method is of little practical interest (being too small).

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):189-195. doi:10.1115/1.3261317.

When two parallel surfaces slide parallel to each other in the presence of a liquid, classical lubrication theory shows that no load carrying ability of the lubricant should result. In experiment after experiment it has been clearly demonstrated that a large and useful load carrying capability often does develop in such situations. Since the successful operation of a significant fraction of sliding bearings and face seals may depend upon this phenomenon, a better understanding would be very useful for bearing and seal design. In this paper much of the known data on parallel sliding experiments are reviewed and compared. Seals as well as bearings are included in this data base. A wide range of conditions and viscosities are included. Some recent work on parallel sliding in water is examined. A comparison of parallel sliding to tilted sliding is also made. It is shown that a strong load support mechanism is present in all of the experimental results. The experiments clearly show that as speed is increased the bearing surfaces are lifted up such that asperity contact and friction are reduced. While explanations are often given for individual sets of experiments, the pervasiveness of this behavior suggests that perhaps there is some mechanism common to all parallel sliding which is not well understood. A mixed friction model is developed and used to explain some of the results. Conclusions are reached concerning several characteristics of parallel sliding. The paper concludes that given the importance of this phenomenon, careful evaluation of the various possible load support mechanisms should be made so that an effective direction for further research can be established.

Commentary by Dr. Valentin Fuster
J. Tribol. 1987;109(1):196-205. doi:10.1115/1.3261319.

When two parallel surfaces slide parallel to each other in the presence of a liquid, classical lubrication theory shows that no load carrying capacity should result. In Part 1 of this work the experimental data showing that such load support does occur are reviewed. In this paper several of the proposed mechanisms for parallel sliding load support are evaluated by combining the mechanism with a mixed friction model and comparing predictions to experimental results. Both water and oil type bearings are considered. While there are exceptions, it has been found that in general the thermal wedge, microasperity lubrication, microasperity collisions, and squeeze effects cannot account for the large apparent load support in the experimental cases studied. However, small deviations from parallel geometry have a strong effect and can easily account for the observed load support, but several questions remain. It is concluded that future investigations should be directed to developing a better understanding of how favorable macrogeometries might be developed and searching for other sources of pressure generation in parallel sliding.

Commentary by Dr. Valentin Fuster

DISCUSSIONS

Commentary by Dr. Valentin Fuster

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