Rotor-Bearing System Modelling

The Rotor-Bearing System Model

A rotordynamic model is a mathematical representation of the rotor-bearing-support system that can be used to compute the speed-dependent modes, the critical speeds and the unbalance response. The model consists of four types of element: the shaft (modelled as beam elements or, in some cases, as solid FE elements), the disks (modelled as lumped mass and inertia elements at specific axial positions), the bearings (modelled as spring-damper elements with possibly speed-dependent and cross-coupled properties), and the supports (modelled as additional spring-damper elements representing the housing flexibility). The model is assembled into a system of equations that includes the gyroscopic terms (from the disk polar inertia and the spin speed), the bearing stiffness and damping, and the shaft bending. The system is then analysed at a series of speeds to produce the Campbell diagram, the critical speeds and the response.

• Shaft: beam elements or solid FE elements — provides bending stiffness and distributed mass
• Disks: lumped mass and inertia elements — provides mass, transverse inertia and polar inertia
• Bearings: spring-damper elements — direct stiffness, cross-coupled stiffness, damping, speed-dependent
• Supports: spring-damper elements — housing flexibility between bearing and ground
• Gyroscopic terms included in the system equations — proportional to disk polar inertia and spin speed

The Beam Rotor Model

The beam rotor model — also called a 1D or Timoshenko beam model — represents the shaft as a series of beam elements along the axis of rotation. Each beam element has bending stiffness (EI), shear stiffness (GAκ for Timoshenko beams), distributed mass and rotary inertia. The disks are attached at specific nodes as lumped elements with mass, transverse inertia (about the diameter) and polar inertia (about the spin axis). The bearings are attached at specific nodes as spring-damper elements. The beam model is efficient — it has few degrees of freedom (two translations and two rotations per node) and it captures the essential physics: shaft bending, disk inertia, gyroscopic coupling and bearing support. For most rotordynamic analyses — Campbell diagrams, critical speeds, unbalance response — the beam model is sufficient. It is the standard approach in rotordynamic software and in engineering practice.

The beam rotor model is highly effective for most rotordynamic analyses. It captures the essential physics — bending, inertia, gyroscopic coupling, bearing support — with few degrees of freedom. It is the standard approach in rotordynamic software and engineering practice.

BEAM MODEL vs 3D FE MODEL

The choice between a beam model and a 3D FE model depends on the complexity of the rotor and the objectives of the analysis. The table below compares the two approaches.

For most rotordynamic analyses — Campbell diagrams, critical speeds, unbalance response — the beam model is sufficient. The 3D FE model is needed for complex geometries, stress concentration and bladed-disk dynamics. The engineer who builds a 3D FE model for a simple shaft-disk system is adding cost without adding value.

The Disk Element

The disk element in a beam rotor model is a lumped element at a specific axial position. It has three inertial properties: the mass (which provides the translational inertia), the transverse moment of inertia (about a diameter — which provides the rotary inertia for bending in each plane), and the polar moment of inertia (about the spin axis — which provides the gyroscopic coupling). The polar inertia is the key property for rotordynamics: it is the source of the angular momentum that produces the gyroscopic moment and the mode splitting. A disk with large polar inertia (a heavy, wide disk) produces strong gyroscopic splitting; a disk with small polar inertia (a light, thin disk) produces weak splitting. The disk element also provides the eccentric mass that creates the unbalance excitation — the mass centre offset from the geometric axis is the source of the synchronous force (Article 02).

• Mass: translational inertia — contributes to the bending mode frequencies
• Transverse inertia (diametral): rotary inertia for bending — adds effective mass to rotational modes
• Polar inertia (axial): source of gyroscopic coupling — determines mode splitting magnitude
• Eccentric mass: source of unbalance excitation — mass centre offset from geometric axis

The Bearing Element

The bearing element in a rotordynamic model is a spring-damper element connecting the shaft node to the support (or to the housing, if the housing flexibility is modelled separately). The element has four properties in each direction: direct stiffness (K_xx, K_yy — force per displacement in the same direction), cross-coupled stiffness (K_xy, K_yx — force per displacement in the perpendicular direction), direct damping (C_xx, C_yy — force per velocity in the same direction), and cross-coupled damping (C_xy, C_yx — usually small, but included for completeness). For a rolling-element bearing, the cross-coupled terms are usually small and the direct stiffness is approximately equal in x and y. For a fluid-film bearing, the cross-coupled terms are significant and the four stiffness coefficients may all be different. The bearing element may be speed-dependent — the stiffness and damping are functions of the rotational speed, and the model uses different values at each speed.

  Bearing element (2D, x-y plane):

  [ F_x ]   [ K_xx  K_xy ] [ x ]   [ C_xx  C_xy ] [ ẋ ]
  [ F_y ] = [ K_yx  K_yy ] [ y ] + [ C_yx  C_yy ] [ ẏ ]

  where  K_ij = stiffness coefficients (force/displacement)
         C_ij = damping coefficients (force/velocity)

  Direct:     K_xx, K_yy — force in same direction as displacement
  Cross-coupled: K_xy, K_yx — force perpendicular to displacement

  Rolling-element: K_xy, K_yx ≈ 0 — approximately axisymmetric
  Fluid-film:     K_xy, K_yx significant — asymmetric pressure distribution

  All coefficients may be speed-dependent.

FE Rotor Models for Complex Geometries

For rotors with complex geometry — bladed disks, impellers, multi-stage compressors with variable cross-sections — the beam model may not capture the local deformation and the complex mode shapes. In these cases, a 3D FE rotor model is used. The shaft, the disks and the blades are modelled with solid (3D) elements, and the full 3D mass and stiffness matrices are assembled. The gyroscopic terms are included by rotating the mass matrix — the Coriolis and centrifugal terms are added to the system equations. The bearing elements are attached at the appropriate surface nodes. The 3D FE model is more expensive — more degrees of freedom, longer computation time — but it captures the local stress and deformation that the beam model cannot. For bladed-disk dynamics (Article 13), the 3D FE model is essential — the blade modes and the disk modes and their coupling cannot be represented by a beam model.

• 3D FE rotor model: solid elements for shaft, disks, blades — full 3D mass and stiffness
• Gyroscopic terms included by rotating the mass matrix — Coriolis and centrifugal terms
• More expensive — more DOF, longer computation — but captures local deformation
• Essential for bladed-disk dynamics — blade and disk modes cannot be represented by beam model

Cross-Link: Engineering Practice and Finite Element Modelling

The rotor-bearing system model is a specialised application of the general finite element method, and it connects to two existing Knowledge Base categories. The Finite Element Modelling category covers the general principles of FE modelling — element selection, meshing, boundary conditions, convergence — that apply to all FE analyses, including rotor models. The Engineering Practice category covers the practical aspects of analysis — model planning, assumption documentation, independent checking, calculation notes — that are essential for a defensible rotordynamic analysis. The engineer who builds a rotor model should follow the general FE modelling principles and the engineering practice principles, in addition to the rotordynamic-specific considerations described in this article.

• Cross-link to Finite Element Modelling: element selection, meshing, boundary conditions
• Cross-link to Engineering Practice: model planning, assumption documentation, independent checking
• The rotor model is a specialised FE model — the general principles apply alongside the rotordynamic-specific ones

Key takeaways

• A 1D beam rotor model — the shaft modelled as beam elements, the disks as lumped masses and inertias, the bearings as spring-damper elements — is highly effective for most rotordynamic analyses. It is not always necessary to build a 3D FE model.
• The disk element provides the mass, the transverse inertia and the polar inertia — the polar inertia is what generates the gyroscopic coupling.
• The bearing element provides the direct stiffness, the cross-coupled stiffness and the damping — each may be speed-dependent.
• The housing flexibility can be included as an additional spring between the bearing and ground, or as part of a coupled FE model if the housing is flexible enough to participate in the rotor modes.
• The model must be exercised across the operating speed range — a single analysis at a fixed speed does not capture the speed-dependent behaviour.