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Research Papers: Hydrodynamic Lubrication

# Elastohydrodynamics Applied to Active Tilting-Pad Journal Bearings

[+] Author and Article Information
Asger M. Haugaard

Department of Mechanical Engineering, Section for Solid Mechanics, Technical University of Denmark, 2800 Lyngby, Denmarkmah@mek.dtu.dk

Ilmar F. Santos

Department of Mechanical Engineering, Section for Solid Mechanics, Technical University of Denmark, 2800 Lyngby, Denmarkifs@mek.dtu.dk

J. Tribol 132(2), 021702 (Mar 25, 2010) (10 pages) doi:10.1115/1.4000721 History: Received March 19, 2009; Revised October 21, 2009; Published March 25, 2010; Online March 25, 2010

## Abstract

The static and dynamic properties of tilting-pad journal bearings with controllable radial oil injection are investigated theoretically. The tilting pads are modeled as flexible structures and their behavior is described using a three-dimensional finite element framework and linear elasticity. The oil film pressure and flow are considered to follow the modified Reynolds equation, which includes the contribution from controllable radial oil injection. The Reynolds equation is solved using a two-dimensional finite element mesh. The rotor is considered to be rigid in terms of shape and size, but lateral movement is permitted. The servovalve flow is governed by a second order ordinary differential equation, where the right hand side is controlled by an electronic input signal. The constitutive flow-pressure relationship of the injection orifices is that of a fully developed laminar velocity profile and the servovalve is introduced into the system of equations by a mass conservation consideration. The Reynolds equation is linearized with respect to displacements and velocities of the nodal degrees of freedom. When all nodal points satisfy static equilibrium, the system of equations is dynamically perturbed and subsequently condensed to a $2×2$ system, keeping only the lateral motion of the rotor. As expected, bearing dynamic coefficients are heavily influenced by the control parameters and pad compliance.

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## Figures

Figure 1

Overview of the different domains, the fluid film curvilinear coordinate system (χ1,χ2), and the Cartesian coordinate system (x1,x2,x3). The figure depicts a generic bearing pad, with an arbitrarily placed orifice. It is to be understood as a qualitative aid.

Figure 2

Overview of the coupled system, with denotion of the equations that govern the different parts, i.e., (2) means that that part is governed by the Reynolds equation. The first coordinates of the Cartesian coordinate system (x1,x2) are shown, the coordinate system is right handed, thus, x3 points out of the plane of the paper. The curvilinear coordinate χ2 is also shown, but is better understood by regarding Fig. 1.

Figure 3

Schematic of the pivot boundary conditions. The pivot is modeled as rigid. Nodes along the surface of the pad are constrained to move tangentially relative to the pivot point.

Figure 4

Visualization of rotor perturbation and result vector components

Figure 5

The finite element mesh used for the computations. Note that the mesh, as shown here, contains roughly 60,000 degrees of freedom. Due to symmetry, only half of these need be considered. The elements are 20 node (second order) serendipity elements. When derived fields (stresses and strains) are not of relevance, this mesh is more than adequate in terms of refinement.

Figure 6

Curves of minimum film thickness for different load cases and pad elastic moduli. High load refers to a radial static bearing load of 20,000 N, and low load corresponds to 2000 N. The data are fitted by polynomials.

Figure 7

Curves of static oil film thickness as a function of χ2 measured from (x1,x2)=(D/2,0) for different injection pressures and pad elastic moduli; (a) E=100 GPa and (b) E=1000 GPa. The bearing is under a static radial load of 20,000 N in the x1 direction, corresponding to S=0.235.

Figure 8

Curves of static pressure at x3=0 as a function of χ2 measured from (x1,x2)=(D/2,0). The bearing is under a static radial load of 20,000 N in the x1 direction, corresponding to S=0.235. Results for the three different values of static injection pressure are shown.

Figure 9

(a) Curves of bearing stiffens for various values of the static injection pressures in the low load case, S=2.35. (b) Curves of bearing stiffens for various values of the static injection pressures in the high load case, S=0.235.

Figure 10

(a) Curves of bearing damping for various values of the static injection pressures in the low load case, S=2.35. (b) Curves of bearing damping for various values of the static injection pressures in the high load case, S=0.235.

Figure 11

(a) Curves of bearing stiffness for proportional control gains in the low load case, S=2.35. (b) Curves of bearing stiffness for proportional control gains in the high load case, S=0.235.

Figure 12

(a) Curves of bearing stiffness for derivative control gains in the low load case, S=2.35. (b) Curves of bearing stiffness for derivative control gains in the high load case, S=0.235.

Figure 13

(a) Curves of bearing damping for proportional control gains in the low load case, S=2.35. (b) Curves of bearing damping for proportional control gains in the high load case, S=0.235.

Figure 14

(a) Curves of bearing damping for derivative control gains in the low load case, S=2.35. (b) Curves of bearing damping for derivative control gains in the high load case, S=0.235.

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