Kxx
x-direction reaction force caused by x displacement.
Bearing Stiffness and Damping Coefficients
How oil-film reaction forces are linearised for critical-speed, unbalance-response and rotor-stability analysis.
Coefficients Describe the Oil Film Around One Operating Point
A journal bearing does not behave like a simple steel spring and dashpot. Its hydrodynamic reaction force changes with shaft position, velocity, speed, load, lubricant condition and bearing geometry. Stiffness and damping coefficients are a local linear model of those force changes for small shaft motions about a calculated equilibrium.
A coefficient value is incomplete unless its operating point, axes, units, force sign, temperature condition and frequency or reduction method are also stated.
Linearised Force Model
From Shaft Motion to Bearing Reaction Force
For small perturbations, a common convention writes the bearing reaction as the negative sum of stiffness, damping and added-mass contributions.
Fb=− ( K q + C q̇ + M q̈ )
Stiffness [K]
KxxKxyKyxKyy
Force per displacement
Damping [C]
CxxCxyCyxCyy
Force per velocity
Added mass [M]
MxxMxyMyxMyy
Force per acceleration
Some software reports applied excitation force rather than bearing reaction force, reverses an axis or uses a different harmonic convention. The numeric signs can therefore differ even when the physical bearing is identical.
x
y
Load
A common horizontal-machine convention uses x as the non-load axis and y as the load axis. Always verify the model definition.
Coefficient Map
How to Read the Subscripts
The first subscript identifies the force direction. The second identifies the motion causing that force.
Kxy
x-direction reaction force caused by y displacement.
Kyx
y-direction reaction force caused by x displacement.
Kyy
y-direction reaction force caused by y displacement.
Cxx
x-direction reaction force caused by x velocity.
Cxy
x-direction reaction force caused by y velocity.
Cyx
y-direction reaction force caused by x velocity.
Cyy
y-direction reaction force caused by y velocity.
Physical Roles
Stiffness, Damping and Cross-Coupling Are Not Interchangeable
Direct Stiffness
Controls how strongly the oil film reacts to displacement in the same axis. It influences bearing-supported natural frequencies, critical speeds, static orientation and mode shapes. Different Kxx and Kyy values indicate directional stiffness or orthotropy.
Direct Damping
Relates reaction force to velocity in the same axis. It influences resonance amplification and how rapidly vibration energy is removed. A larger C value does not automatically mean more effective system damping because mode shape and stiffness also matter.
Cross-Coupled Terms
Connect motion in one axis with force in the other. Their signs and relative magnitudes can add or remove energy from a forward whirl. Tilting-pad bearings commonly reduce destabilising stiffness cross-coupling, but it must not be assumed to be exactly zero.
Units and Terminology
Keep Dimensional and Normalised Data Separate
| Quantity | Common SI unit | Common scaled form | Meaning |
|---|---|---|---|
| Stiffness K | N/m | MN/m | Incremental reaction force per unit displacement. |
| Damping C | N·s/m | kN·s/m | Incremental reaction force per unit relative velocity. |
| Added or virtual mass M | kg | kg | Incremental reaction force per unit relative acceleration. |
| Dynamic stiffness / impedance | N/m | MN/m, complex | Frequency-domain force-to-displacement relation combining K, C and M. |
| Dimensionless coefficients | No universal unit | Normalised | Scaled by a stated load, clearance, speed or geometry convention. The normalisation definition is essential. |
Z(ω) = K − ω2M + iωC
One common harmonic convention for dynamic stiffness. The sign of the imaginary term changes if the assumed time dependence changes, so the convention must accompany complex data.
Operating Conditions
Why the Coefficients Change
A bearing coefficient table is a map of operating points, not a permanent nameplate property.
Speed and Load
Journal speed, load magnitude and load angle determine equilibrium position and hydrodynamic pressure distribution.
Oil and Temperature
Viscosity, inlet temperature, hot-oil carryover, flow regime and supply arrangement change film force and damping.
Clearance and Preload
Hot assembled clearance, pad curvature and journal bearing preload strongly affect stiffness and damping.
Pad and Pivot Flexibility
Pad bending, pivot contact stiffness and carrier or housing compliance can sit in series with the oil-film stiffness.
Bearing Configuration
Pad count, L/D ratio, pivot offset, load-on-pad or load-between-pad orientation and lubrication method all matter.
Perturbation Frequency
Excitation frequency, synchronous reduction and any retained pad degrees of freedom affect the coefficients supplied to the rotor model.
Frequency Dependence
When [K][C] Is Not the Whole Model
Tilting-pad bearings introduce pad motion, pivot flexibility and fluid inertia. Depending on the formulation, their journal-level coefficients may vary with perturbation frequency.
Texas A&M and GE testing reviewed by the Turbomachinery Laboratory found that direct real dynamic stiffness could often be represented as a quadratic function of frequency, with direct damping represented as approximately constant, by adding an [M] matrix to form a frequency-independent [K][C][M] model.
Basic model
[K] [C]
Suitable when the supplied coefficients and analysis method support a frequency-independent stiffness and damping representation.
Extended model
[K] [C] [M]
Adds virtual or added mass to represent part of the measured frequency dependence without changing K and C at every frequency.
Minimum Data Set
What Must Accompany the Coefficient Matrix
01
Shaft speed, static bearing load and load angle
02
x/y axis orientation and load-on-pad or load-between-pad arrangement
03
Dimensional units and any coefficient normalisation
04
Reaction-force sign and matrix subscript convention
05
Oil grade or viscosity, inlet temperature and lubrication condition
06
Hot clearance, preload and equilibrium shaft position
07
Perturbation frequency, synchronous reduction or [K][C][M] formulation
08
Thermal, pad, pivot, housing and turbulence assumptions used in the model
Engineering Workflow
How Coefficients Become a Rotor Model
- 1
Solve Steady State
Establish equilibrium position, film thickness, pressure, temperature, flow and pad attitude at each operating point.
- 2
Perturb the Journal
Apply small displacement and velocity perturbations, or a frequency-domain excitation, about that equilibrium.
- 3
Extract Coefficients
Linearise the reaction forces into direct and cross-coupled K, C and, where required, M matrices.
- 4
Assemble the Rotor
Insert bearing matrices at the correct stations and coordinates with shaft, disks, seals, couplings and supports.
- 5
Assess the System
Calculate critical speeds, mode shapes, unbalance response, separation margins and stability over the operating range.
Common Errors
Why Coefficient Comparisons Go Wrong
Comparing coefficients at different speed, load, oil temperature or clearance.
Swapping Kxy and Kyx or rotating axes without transforming the matrix.
Mixing N/m with MN/m, or dimensional values with normalised coefficients.
Assuming a positive damping number guarantees adequate rotor stability.
Using cold geometry when the coefficient model assumes hot operating clearance.
Ignoring frequency reduction, added mass, pivot stiffness or pad flexibility.
Common Questions
Stiffness and Damping FAQ
Are coefficients material properties?
No. They describe the complete operating oil film and bearing structure about one equilibrium point. Lining, pad and pivot materials can influence them, but do not define them alone.
Why are Kxx and Kyy different?
Load direction, shaft eccentricity, pad orientation and geometry make the bearing directionally asymmetric. This is commonly described as stiffness orthotropy.
Is more damping always better?
Not as a standalone coefficient. Effective modal damping depends on stiffness, mode shape, cross-coupling and the complete rotor-bearing system.
Can one coefficient set cover every speed?
Normally no. Coefficients should be calculated across the operating and transient range required by the rotor-dynamic study.
Technical Basis
The force model and coefficient definitions were cross-checked against Texas A&M Turbomachinery Laboratory research on tilting-pad bearing force measurements, measured frequency characteristics, pad and pivot flexibility and the tutorial Fundamentals of Fluid Film Journal Bearing Operation and Modeling. This guide explains interpretation and does not provide machine-specific coefficients.
Need Bearing Coefficients for a Rotor Model?
Send the bearing geometry, shaft data, operating envelope, lubricant conditions and rotor-dynamic requirements for an engineering review.