A theoretical basis for static and dynamic operation of tilting pad journal bearings (TPJBs) has evolved over the last 50 years. Originally demonstrated by Lund using the pad assembly method and a classic Reynolds equation solution, the current state of the art includes full thermoelastohydrodynamic solutions of the generalized Reynolds equation that include fluid convective inertia effects, pad motions; and thermal and mechanical deformations of the pads and shaft. The development of TPJB theory is reviewed, emphasizing dynamic modeling. The paper begins with the early analyses of fixed geometry bearings and continues to modern analyses that include pad motion and stiffness and damping effects. The development of thermohydrodynamic, thermoelastohydrodynamic, and bulk-flow analyses is reviewed. The theories of TPJB dynamics, including synchronous and nonsynchronous models, are reviewed. A discussion of temporal inertia effects in tilting pad bearing is considered. Future trends are discussed, and a path for experimental verification is proposed.

Rotating machinery such as pumps, compressors, fans, turbines, and generators are ubiquitous in industrial settings. Fluid film journal bearings are suited to applications experiencing higher speeds and loads, often in excess of 90 m/s surface velocity and with typical bearing specific loads of 700 kPa–3.5 MPa. In these bearings, hydrodynamic action is used to support the rotor on a thin lubricating film. Typical film thicknesses between the rotor and the bearing surface are on the order of 100

The hydrodynamic action in fluid film bearings is fundamentally a fluid-structure interaction effect. When these effects are linearized and perturbed in the two orthogonal radial directions relative to the shaft, they result in equivalent lateral stiffness and damping coefficients which can then be used in lateral vibration analysis of rotors.

Tilting pad journal bearings are a source of both static support and dynamic stiffness and damping. Tilting pad journal bearings have a number of pads, typically four or five. A four pad bearing is shown in Figure

Tilting pad journal bearing.

High-speed rotordynamic applications often have rotors that pass through one or two bending critical speeds as the machines are accelerated to the operating speed. The damping from the fluid film bearings is required to safely pass through these bending critical speeds as the rotating element is accelerated. The damping also helps suppress potentially destabilizing forces from sources such as radial seals, balance pistons, impeller eye seals, internal friction fits, and unbalanced electromagnetic forces [

Characterization of the dynamic response of tilting pad bearings is vital to successful design of high-speed rotating machinery. Theoretical models exist for prediction of the dynamic response. These are particularly important at the design stage of modern high-speed rotating machinery. These models have evolved from analytical solutions of the lubricating film of fixed geometry bearings to full finite element and finite difference numerical solutions that include analysis of the lubricating flow; the energy balance between the lubricant, the bearing, and the rotor; and mechanical and thermal deformations of the shaft and bearing pads.

Additionally, there is still some controversy within the rotordynamic community over the proper dynamic modeling method for tilting pad journal bearings. Some researchers question whether consideration of excitation frequencies other than rotor operating speed is necessary. There is also continued discussion on the number of degrees of freedom to retain within the bearing, either implicitly or explicitly, when nonsynchronous dynamic models are considered. The issue of the presence of fluid temporal inertia effects also arises as part of these discussions. This paper will review and discuss those issues.

This paper is organized into nine sections. In Section

Many of the investigators of bearing dynamics were concerned with the onset of rotordynamic instability, so stability assessments in the literature are common. Less common are predictions of critical speeds and unbalance response. The previous work considered in this paper is not comprehensive, but is representative of developments of bearing models. The works cited in this paper and their references do give a comprehensive treatment of bearing modeling developments.

Section

Section

Section

Discussion and conclusions are provided in Section

The fundamental lubrication equation was originally formulated by Reynolds in 1886 [

Equation (

Plain slider.

The Sommerfeld number recognizes the effect of the net force applied to the shaft at the bearing,

One of the assumptions in arriving at the Sommerfeld number is that the lubricating flow is laminar. While appropriate for lower surface speed bearings where the laminar flow assumption can be justified, the assumption is increasingly violated due to the high rotational speeds typical for many modern bearings. An additional dimensionless group for evaluating bearings operating in the turbulent regime is the Reynolds number using the bearing diametral clearance

The analyses of Reynolds and Sommerfeld focused solely on the solution to the hydrodynamic flow field problem, described by (

Other bearing design geometric properties will enter the discussion. The dynamic properties of bearings are affected by the bearing design geometry. The geometric properties are as follows.

Bearing preload,

Pivot location: pivot location relative to the leading edge of the pad expressed as a percentage of pad arc length

Load orientation: napplied load relative to the bearing pads. Load on pad and load between pad configurations are typical.

The solutions by Reynolds and Sommerfeld were for the pressure field and the net forces of the lubricating oil film. Most analysts of the era considered the rotor to be simply supported at the bearings. As the understanding of the linearized bearing response improved, researchers recognized the equivalent stiffness and damping effects provided by the lubricating film.

The first attempts to quantify the dynamic response of the lubricating film itself were made by Stodola [

Another early analysis that recognized the effect of bearing flexibility on critical speeds was reported by Linn and Prohl [

Fixed geometry radial bearings were standard in the first half of the 20th century, and tilting pad bearings only saw significant adoption begin during the 1960s. However, the tilting pad thrust bearing was invented independently by Kingsbury and Michell. Michell also invented the tilting pad journal bearing and installations of the tilting pad journal bearing appear as early as 1916 [

Michell Combined Tilting Pad Journal and Thrust Bearing [

“[T]he plain journal bearing compares favorably with the pivoted-pad bearing and by many criteria is somewhat superior to the latter.”

Because of these factors, the perceived drawbacks to tilting pad bearings outweighed the benefits.

The advantages of tilting pad bearings in removing the bearings as a source of self-excited vibrations was originally recognized by Hagg in 1946 [

While the development reported by Hagg was significant, many analysts continued to work with plain journal bearings. The benefit of improved stability margin was still not significant enough to designers of the era to overcome the perceived drawbacks discussed previously. Additionally, noncircular bearing bores were discovered to enhance the stability margin and were a lower cost option to a tilting pad arrangement.

Sternlicht [

Solutions to the perturbed Reynolds equation also began to appear in textbooks, including the ones by Smith [

Even with the improvements to fixed geometry designs to enhance stability, there is still a limit where the destabilizing forces are high enough to overcome the damping and drive the rotor unstable. Typically, the limit is reached when the operating frequency is greater than twice the first bending natural frequency [

Adoption of tilting pad bearings was also made easier by analytical solutions that allowed designers to understand the dynamic properties. One of the major advances in understanding the dynamics of tilting pad bearings came from Lund's landmark paper in 1964 [

The landmark work by Lund led to a significant research effort to extend the analyses of tilting pad bearings. Thermohydrodynamic and TEHD solutions and turbulence corrections for high rotational speeds also begin to appear for tilting pad bearings. The use of synchronously reduced coefficients to described the linearized dynamics became the norm, following the results reported by Lund [

Orcutt [

Nicholas et al. [

Nicholas and Kirk [

Jones and Martin [

Ettles [

Hashimoto et al. [

Knight and Barrett [

Brugier and Pascal [

Ettles [

Brockwell et al. [

Hopf and Schüeler [

Hyun et al. [

Nicholas and Wygant [

Several researchers have investigated transient effects in tilting pad bearings. These studies were influenced in part by a review of bearing failures presented by Conway-Jones and Leopard [

Transient effects were also considered by Monmousseau et al. [

The work by Lund in 1964 [

Shapiro and Colsher [

Allaire et al. [

Parsell et al. [

Rouch [

Lund and Pedersen [

Branagan [

Barrett et al. [

Earles et al. [

White and Chan [

Brockett and Barrett [

Kim et al. [

Tilting pad bearing lubrication theory has evolved, from fixed geometry isoviscous analytical solutions, to advanced finite element solutions including hydrodynamic, energy, and deformation effects. The modifications to (

Modern tilting pad bearing lubrication theory is based on thermoelastohydrodynamic models that include equations describing the hydrodynamic flows, heat transfer and shear heating, and mechanical deformations [

Reynolds' equation, (

The effective cross-film viscosities

The flow profile in the bearing is treated as a combination of Couette and Poiseuille flow, which is expressed as:

The solutions to (

The viscosity of many lubricants is a strong function of temperature. The developed pressures in hydrodynamic bearings are not large enough to significantly affect the viscosity. To account for temperature effects, the 2D energy equation, including shear heating terms, is considered in the model presented in [

The turbulence model implemented by He [

Equations (

Once the pressure profile solution is found, the generalized Reynolds equation is perturbed. The first-order perturbation results in the equivalent stiffness and damping coefficients,

When (

Another approach to the lubrication problem is the averaged flow method, where the properties of the lubricant across the film are averaged. First proposed by Constantinescu [

Other authors have used predominantly empirical bulk flow approaches, including Hirs [

The averaged flow approaches rely on averaging of the fluid properties across the film, including average velocity and viscosity. For example, the formulations typically consider an average fluid velocity of the form:

Section

Constantinescu published a series of papers [

The series of papers culminated in a journal bearing lubrication theory published by Constantinescu and Galetuse in 1982 [

For laminar flows,

Hirs [

By solely considering the average flow properties and the boundary conditions, Hirs developed a set of pressure equations for sliding surfaces as

In terms of friction factors, the Hirs approach can generally produce more accurate results for the lubrication transition flow regime because experimental data is fitted to the model, assuming such data is available for that bearing configuration and flow condition. The transition region is where many modern oil-lubricated bearings operate. The key drawback is that the method totally relies on empirical data. The types of experiments that would be required to obtain a complete set of empirical coefficients were alluded to by Hirs [

Taylor and Dowson [

The transition region from laminar flow to turbulent flow presents challenges to the both the eddy-viscosity model and the bulk flow approaches. Suganami and Szeri [

Bouard et al. [

The tilting pad dynamics developed from various TEHD models are based on explicit modeling of the motion of the pads. The modeling procedure is summarized in the following section. The development of the bearing model reduced to the shaft degrees of freedom and an experimentally identified two-degree-of-freedom bearing model are also summarized.

The lubricating film is typically represented with stiffness and damping coefficients in linear analyses. To illustrate this concept, two free-body diagrams are provided. The first, Figure

Free body diagram, dhaft translational degrees of freedom, and rigid pivots.

The second free-body diagram, Figure

Free body diagram, pad rotational degrees of freedom, and rigid pivots.

When a force balance is considered on the free body diagrams, the resulting equations of motion can be expressed in matrix form as [

Once the individual pad equations of motion are transformed to global coordinates, the overall equations of motion can be assembled [

Equation (

There were also some fundamental misunderstandings of the tilting pad journal bearing results originally presented by Lund in 1964 [

For the purposes of this discussion, (

Dynamic reduction is performed in the frequency domain by assuming a solution of the form

If the perturbation frequency

The reduced-order model with pad dynamics considered implicitly, (

An alternative experimental approach to characterizing TPJB behavior is based on an experimentally identified model in the frequency domain. The experimentally derived model is based on measurement of force inputs and bearing housing outputs. The shaft is held rigidly in rolling element bearings, and the bearing is allowed to move radially. The bearing housing is perturbed and displacements of the bearing relative to the shaft are measured. The method, originally applied to fixed pad hydrostatic bearings [

The system identification method employed in [

The real and imaginary parts of the complex impedance functions

These results were reviewed by Childs [

When the frequency response data for flexible pivot bearings reported in [

The inclusion of temporal inertia effects in the bulk flow model is justified in [

The theory proposed by San Andrés is in contrast to the TEHD analysis based on the generalized Reynolds equation proposed by He [

A discussion of the effect on stability analysis is also presented in [

There have been two distinct approaches to temporal inertia effects in hydrodynamic lubrication documented in the literature. The two approaches were compared originally in [

Both Reinhardt and Lund [

Both analyses agree on the nondimensional form of the perturbed N-S equations with inertia terms. The general approach to calculate rotordynamic coefficients is to perform a Taylor series expansion about the reduced Reynolds number, resulting in an

Both analyses also essentially agree on the nondimensionalization of force, damping coefficients, and stiffness coefficients, with minor differences in expression of rotational speed:

When considering a laminar lubricating type flow, the flow results are dominated by fluid shear effects. As a result, the fluid viscosity is more fundamental than fluid density in nondimensionalizing and scaling the results. This would imply that for low reduced Reynolds number, the fluid inertia (added mass) effects are not significant. Using data derived in [

The Taylor series expansion resulting in (

Temporal inertia terms will be important for bearings with low viscosity lubricants or high

Since the original development of the lubrication equation by Reynolds [

Generally, consideration of more complex lubrication models followed the experience of industrial users. As the classical Reynolds solution diverged from user experience, the addition of energy and deformation effects into the analysis became necessary to allow for reliable designs. The factors requiring these modeling improvements, including increasing speeds and bearing specific loads, are demands by industrial users that continue to influence the need to improve tilting pad bearing models.

There has been a similar evolution in the understanding of the bearing dynamics, especially with tilting pad journal bearings. Initially treated as simple supports, inclusion of stiffness effects and later damping effects improved the understanding of the bearing contribution to the overall rotordynamic system. These improvements came as user experience did not match with simpler bearing dynamic models.

Tilting pad bearings were adopted to address self-excited vibrations from the fluid structure interactions within fixed pad bearings. The initial understanding was that the synchronous response was a sufficient representation of the bearing dynamics regardless of excitation frequency based on a misinterpretation of the work by Lund. More recent investigations, especially into rotordynamic stability, indicate that the dynamic response is excitation-frequency-dependent.

The nonsynchronous modeling presented in Sections

The KCM experimentally identified model is a fundamentally different model compared to the full bearing coefficient model. The full bearing coefficients are obtained from first principles. The KCM model is based solely on system identification experiments and arises from observation of the system. The observations are consistent with a 12-coefficient second-order nonsynchronous dynamic representation with frequency-independent stiffness, damping, and mass coefficients. This is a “black box” identification technique this is suitable for obtaining a tentative system model and is a technique that is also popular in the controls community for developing an approximate model of a plant to be controlled.

The issue of the relative importance of temporal inertia effects is still an open area of discussion. There are conflicting treatments in the literature of the relative importance of the temporal inertia term in the Navier-Stokes equations, and the assumptions made in developing these treatments are being invalidated by current and projected operating speeds and loads in industrial bearings. Development of a new approach to the generalized Reynolds equation or another simplified form of the Navier-Stokes equations is an opportunity for future research.

The proper dynamic model for tilting pad journal bearings is another area of research and discussion in the literature. This paper summarizes the two approaches, and new methods have been developed to directly compare the two approaches [

Once the coordinate transformation is applied, the single-pad stiffness matrix for a tilting pad bearing with rigid pivots is

Once the coordinate transformation is applied, the single-pad damping matrix for a tilting pad bearing with rigid pivots is