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Technical Brief

# Mobility/Impedance Methods: A Guide for Application

[+] Author and Article Information
J. F. Booker

Professor Emeritus
Fellow ASME, IMechE
Sibley School of Mechanical and Aerospace Engineering,
Cornell University,
Ithaca, NY 14853

The former involves only algebraic adjuncts to low-order ordinary differential equations (ODEs), while the latter involves partial differential equations (PDEs) with free boundaries.

While physical experimentation is attractive in principle, it is rarely done in practice [1,2].

A further step of spatial integration is required to produce data for the related impulse method of Block [3]. Other bearing performance data (e.g., pressure, flow, etc.) can be stored similarly [4].

The concept(s) can be applied to other geometries as well [2], but the main application by far is to plain full journal bearings.

While the mobility/impedance data are normally described in vector terms, it is also possible to treat them in tensor form [8].

While the methods formally require perfect angular alignment, they can also be applied approximately to cases with angular misalignment [10].

While the methods formally require circumferential symmetry, they can also be applied approximately to cases without it (e.g., bearings with incomplete circumferential grooves, inlet holes, partial arcs, etc.) so long as the loaded region has circumferential symmetry; in fact, they can be applied exactly to such bearings if the load direction is fixed relative to the bearing [2].

Vector e is eccentricity of journal relative to sleeve. Vector VS is (squeeze) velocity of journal relative to sleeve. Vector FL is film force (load) applied by journal to sleeve.

Subscripts 1, 2 refer to coordinate systems. Subscripts S, L here (and E elsewhere) identify specific velocity and force vectors as defined in the Nomenclature section; they are consistent with usage in earlier published work by others [7].

In Fig. 3(a) coordinate X1 is aligned with vector M. In Fig. 3(b), coordinate X2 is aligned with vector W.

For example, εX1, εY1 and εX2, εY2 are components of eccentricity ratio vector ε in special coordinate systems X1,Y1,Z1 and X2,Y2,Z2, respectively; they are related by a coordinate rotation through relative attitude angle ψ.

Film force FL is negative here because it is applied to the sleeve (not the journal).

Higher-order integration methods, often implicit, are commonly used for such problems as this.

Time integration is 4th-order Runge–Kutta with fixed time step corresponding to 0.1 deg of shaft rotation.

Higher-order integration methods are commonly used for such problems as this.

Initial transient response is not shown.

Time integration is 2nd-order Runge–Kutta with fixed time step corresponding to 0.5 deg of crankshaft rotation. Results for 4th-order Runge–Kutta integration are virtually identical; results for 1st-order Runge–Kutta (Euler) integration are similar.

A “finite bearing” (complete) solution satisfies the full Reynolds PDE. A “short bearing” (approximate) solution satisfies a Reynolds PDE modified so as to neglect all circumferential flow in a nonrotating bearing. A “long bearing” (approximate) solution satisfies a Reynolds PDE modified so as to neglect all axial flow.

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received July 7, 2013; final manuscript received September 29, 2013; published online December 18, 2013. Assoc. Editor: Daniel Nélias.

J. Tribol 136(2), 024501 (Dec 27, 2013) (8 pages) Paper No: TRIB-13-1134; doi: 10.1115/1.4025760 History: Received July 07, 2013; Revised September 29, 2013

## Abstract

Mobility and impedance methods provide extremely efficient and robust journal orbit/trajectory calculation through use of bearing characteristics stored (at least conceptually) in the form of widely-available “maps.” These inversely-related methods are widely used for design optimization and for dynamic system analysis, both of which are often computationally intensive. All concepts and data sources necessary for straightforward application to both rotating and reciprocating machinery are identified and presented in some detail in this brief note.

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## References

Campbell, J., Love, P. P., Martin, F. A., and Rafique, S. O., 1967–68, “Bearings for Reciprocating Machinery: A Review of the Present State of Theoretical, Experimental, and Service Knowledge,” Proc. Inst. Mech. Eng., Pt. 3A, 182, pp. 51–74, 129–131, 140, 145–146, 148–149, 610–613.
ten Napel, W. E., Moes, H., and Bosma, R., 1976, “Dynamically Loaded Pivoted Pad Journal Bearings: Mobility Method of Solution,” ASME J. Lubr. Technol., 98(2), pp 196–205, 229.
Blok, H., 1975, “Full Journal Bearings Under Dynamic Duty: Impulse Method of Solution and Flapping Action,” ASME J. Lubr. Technol., 97(2), pp. 168–179.
Booker, J. F., 1969, “Dynamically-Loaded Journal Bearings: Maximum Film Pressure,” ASME J. Lubr. Technol., 91(3), pp. 534–543.
Booker, J. F., 1965, “Dynamically-Loaded Journal Bearings: Mobility Method of Solution,” ASME J. Basic Eng., 87(3), pp. 537–546.
Booker, J. F., 1971, “Dynamically-Loaded Journal Bearings: Numerical Application of the Mobility Method,” ASME J. Lubr. Technol., 93(1–2), pp 168–176, 315.
Childs, D., Moes, H., and van Leeuwen, H., 1977, “Journal Bearing Impedance Descriptions for Rotordynamic Applications,” ASME J. Lubr. Technol., 99(2), pp. 198–214.
Moes, H., Sikkes, E. G., and Bosma, R., 1976, “Mobility and Impedance Tensor Methods for Full and Partial-Arc Journal Bearings,” ASME J. Lubr. Technol., 108(4), pp. 612–619.
Booker, J. F., 1989, “Basic Equations for Fluid Films With Variable Properties,” ASME J. Tribol., 111(3), pp. 475–483.
Boedo, S., 2013, “A Hybrid Mobility Solution Approach for Dynamically Loaded Misaligned Journal Bearings,” ASME J. Tribol., 135(2), p. 024501.
Boedo, S., and Booker, J. F., 2009, “Dynamics of Offset Journal Bearings—Revisited,” IMechE J. Eng. Trbol., 223(3), pp 359–369, 606–607.
Goenka, P. K., 1984, “Analytical Curve Fits for Solution Parameters of Dynamically Loaded Journal Bearings,” ASME J. Tribol., 106(4), pp. 421–428.
Childs, D., 1993, Turbomachinery Rotordynamics, Wiley, New York.
Booker, J. F., 1965, “A Table of the Journal Bearing Integral,” ASME J. Basic Eng., 87(2), pp. 533–535.
Booker, J. F., 1984, “Squeeze Films and Bearing Dynamics,” CRC Handbook of Lubrication, Volume II: Theory and Design, E. R.Booser, ed., CRC Press, Boca Raton, FL, pp. 121–137.
Moes, H., and Bosma, R., 1981, “Mobility and Impedance Definitions for Plain Journal Bearings,” ASME J. Lubr. Technol., 103(3), pp. 468–470.
Moes, H., 2000, Lubrication and Beyond, (UT Lecture Notes No. 115531), University of Twente, Enschede, Netherlands.

## Figures

Fig. 2

Special coordinate systems

Fig. 3

Normalized clearance space: (a) impedance, (b) mobility

Fig. 1

Nonrotating journal bearing: geometry

Fig. 4

“Finite bearing” vector fields (L/D = 1/4): (a) impedance, (b) mobility

Fig. 5

“Finite bearing” vector fields (L/D = 1/4): (a) impedance, (b) mobility

Fig. 6

“Finite bearing” orbit/trajectory plots: (a) impedance method, rotating machinery; (b) mobility method, reciprocating machinery

Fig. 7

“Short bearing” vector fields (L/D = 1): (a) impedance, (b) mobility

Fig. 8

“Long bearing” vector fields (L/D = ∞): (a) impedance, (b) mobility

## Errata

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