Technology / Technical guide
Bearing Stiffness and Damping Coefficients
A bearing reaction changes when the shaft moves. Dynamic coefficients describe that small change around a chosen state; they are not a single universal property printed beside a load rating.
From a support reaction to a local dynamic model.
The useful question is how force changes with displacement and velocity, in defined axes and under stated conditions. A matrix can capture effects that one scalar spring cannot.
01 / Local equilibrium
Why the operating point comes first.
A support carrying a steady reaction has an equilibrium shaft position.
A support carrying a steady reaction has an equilibrium shaft position. Write a small displacement from it as δx, rather than treating every motion as measured from an unloaded geometric centre. The steady force and the changing force then have different roles in the equations.
Consider a nonlinear one-dimensional reaction with a local force-displacement slope. Two different equilibrium positions can have different slopes even though they belong to the same bearing. A coefficient evaluated at one state is a tangent approximation, not the complete force law. This differs from the geometric state explained in bearing clearance and preload.
02 / Force convention
Define exactly which force the matrix describes.
For force exerted by the support on the shaft, choose the convention δF = −Kδx −Cδv.
For force exerted by the support on the shaft, choose the convention δF = −Kδx −Cδv. Here δx is a displacement vector, δv its time derivative, and K and C are local coefficient matrices. Reversing the reported force direction changes the signs used in its definition.
Translational stiffness has units N/m and translational damping N·s/m. Multiplication by metres or metres per second produces newtons in each case. Angular coordinates require their own torque-based units; putting them into the same matrix without clear coordinate scales can hide a dimensional error.
03 / Cross-coupling
A displacement can change force in another direction.
In two transverse axes, the x-force can contain both Kxxδx and Kxyδy.
In two transverse axes, the x-force can contain both Kxxδx and Kxyδy. Kxy is therefore not a second x-direction spring: it describes x-force sensitivity to y-motion under the stated convention. The corresponding Kyx term answers a different directional question.
For a hypothetical Kxy of 300 N/m, a y-displacement of 0.002 m contributes −0.6 N to x-force in this convention. Other terms may add to or oppose that contribution. This example illustrates matrix indexing, not a measured bearing coefficient or a prediction of shaft displacement.
04 / Energy relation
Not every stiffness matrix represents stored energy.
A symmetric stiffness matrix can be associated with a quadratic elastic energy, E = δxᵀKδx/2, when the support is conservative and the coordinates are consistent.
A symmetric stiffness matrix can be associated with a quadratic elastic energy, E = δxᵀKδx/2, when the support is conservative and the coordinates are consistent. A general nonsymmetric matrix cannot automatically be interpreted as that same stored-energy model.
Likewise, the damping contribution has instantaneous mechanical power −δvᵀCδv. Its sign depends on the symmetric part of C; a coefficient called damping does not by its name prove positive energy dissipation in every direction. Coupling and the surrounding system determine the response.
05 / System dynamics
Bearing coefficients are inputs, not rotor modes.
A rotor equation combines inertia, shaft and support stiffness, damping, and any relevant rotation-dependent terms.
A rotor equation combines inertia, shaft and support stiffness, damping, and any relevant rotation-dependent terms. Changing one bearing coefficient changes that assembled problem, but it does not directly specify a new natural frequency. A support may be stiff locally while another part dominates flexibility.
A Campbell diagram describes modal frequencies against rotation speed. It is downstream of a complete model. Comparing two such diagrams requires consistent support coordinates and coefficient conditions, rather than changing an isolated number while silently changing the model boundary.
06 / Model boundary
Different bearing mechanisms need different coefficient laws.
COMSOL’s journal-bearing example illustrates coefficients evaluated about equilibrium through a perturbed fluid-film model.
COMSOL’s journal-bearing example illustrates coefficients evaluated about equilibrium through a perturbed fluid-film model. Its hydrodynamic geometry and numerical values are specific to that example. They do not establish the construction of a wind turbine bearing.
Rolling contact, fluid films, housing flexibility and other support mechanisms need their own appropriate descriptions. Large movement, contact changes or frequency-dependent behaviour can exceed a simple constant-coefficient approximation. No coefficient matrix or bearing construction for a MARSTON model is inferred here.
Related technical guides: Campbell Diagrams: Modal Branches, Rotor Orders & Crossings · Rotor Static Unbalance & Couple Unbalance: Force and Moment · Two-Inertia Torsional Vibration in Wind Generators.
General background: COMSOL — Dynamic coefficients of a cylindrical journal bearing. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Equilibrium | Reference position and steady reaction | The origin of the incremental model |
| Direct stiffness | Kxx in N/m | Same-axis force sensitivity |
| Cross stiffness | Kxy in N/m | x-force change caused by y-motion |
| Direct damping | Cxx in N·s/m | Velocity-dependent force sensitivity |
| Model validity | Local conditions and coordinate convention | Coefficients are not universal load ratings |
Engineering questions.
Is bearing stiffness the same as its load rating?
No. A load rating and an incremental force-displacement coefficient describe different quantities.
What does cross-coupled stiffness mean?
Motion in one coordinate changes force in another. The force direction, motion direction and sign convention must be identified.
Are Kxy and Kyx always equal?
No. Equality follows for certain conservative models, but it should not be imposed on every support mechanism.
Does a positive diagonal damping term prove stability?
No. Other matrix terms and the complete rotor system also affect energy and stability.
Can one set of coefficients describe every operating state?
Only within an appropriate approximation. Changes in equilibrium or physical conditions can change the local model.
Do these coefficients describe MARSTON bearings?
No. The page explains support modelling without claiming a specific bearing type or measured coefficient set.
Technical enquiry
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A bearing reaction changes when the shaft moves. Share the model and applicable operating or design information for an engineering discussion.