Dynamic Analysis of Rotating Structures
Dynamic Analysis of Rotating Structures: Theoretical Foundations
Special Characteristics of Rotating Body Vibration
Professor, how is the vibration of a rotating body different from that of a stationary structure?
Rotating bodies have three special effects:
1. Stiffness change due to centrifugal force (Spin Stiffening) — Handled in prestressed modal analysis
2. Coriolis Force — A fictitious force acting on vibrating objects when viewed from a rotating coordinate system
3. Gyroscopic Effect — An effect that occurs when the orientation of a rotating object changes
So Coriolis force and gyroscopic effect are added.
Equation of motion:
- $[G]$ — Gyroscopic matrix (Antisymmetric. Force proportional to velocity)
- $[K_\sigma]$ — Geometric stiffness (Spin stiffening due to centrifugal force)
- $[K_c]$ — Centrifugal softening (spin softening)
$[G]$ is antisymmetric! It goes in the same position as the damping matrix $[C]$, but it's asymmetric, right?
$[G]$ is antisymmetric ($[G]^T = -[G]$) and neither adds nor removes energy from the system. However, it causes mode splitting (the phenomenon where forward and backward whirl frequencies differ).
Campbell Diagram
The Campbell Diagram, which plots rotational speed $\Omega$ vs. Natural Frequency $f$, is a fundamental tool for rotating body vibration analysis.
Characteristics:
- Forward Whirl — Vibration in the same direction as rotation. Frequency increases as $\Omega$ increases.
- Backward Whirl — Vibration opposite to the rotation direction. Frequency decreases as $\Omega$ increases.
- Excitation Lines ($f = n\Omega$) — Intersections with unbalance ($n=1$) or higher-order excitations indicate resonance.
Forward and backward whirls split... is this the gyroscopic effect?
Yes. Two modes that had the same frequency in a stationary state split as the rotational speed increases. Accurate prediction of this splitting is necessary in the design of turbine blades and rotors.
Critical Speed
Critical speed corresponds to the rotational speed at the intersection of the excitation line and the natural frequency line. The intersection with unbalance excitation ($n=1$) is particularly important.
Is the basic principle to avoid operating at critical speeds?
Basically, yes. Standards like API 617 (Compressors) and API 612 (Steam Turbines) set a margin of ±15% from the critical speed.
Summary
Let me organize the vibration of rotating bodies.
Key Points:
- Three Special Effects — Centrifugal stiffness change, Coriolis Force, Gyroscopic Effect
- Gyroscopic matrix $[G]$ is antisymmetric — Causes forward/backward whirl mode splitting
- Campbell Diagram — Rotational speed vs. Natural Frequency. Identifies resonance conditions.
- Critical Speed — Intersection with excitation lines. Avoid with ±15% margin.
- Complex Eigenvalue Analysis is Required — Eigenvalues become complex due to gyroscopic effect.
History of the Discovery and Application of the Gyroscopic Effect
The gyroscopic effect was discovered in 1852 by the French physicist Léon Foucault while studying the stability of a top. He also named it the gyroscope. The property of a rotating body's angular momentum resisting moments was applied to radio compasses in the early 1900s and remains a core technology in INS (Inertial Navigation Systems) today. In FEM, the gyroscopic effect is expressed as an antisymmetric gyroscopic matrix [G].
Computational Methods for Dynamic Analysis of Rotating Structures
FEM Analysis of Rotating Bodies
How do you perform rotating body vibration analysis with FEM?
Nastran
```
SOL 107 $ Complex eigenvalues (including gyroscopic effect)
CEND
CMETHOD = 10
BEGIN BULK
RFORCE, 100, 1, , 100., 0., 0., 1. $ Rotational speed 100 rad/s, rotation about z-axis
```
SOL 107 for complex eigenvalue analysis including gyroscopic matrix. RFORCE specifies rotational speed and axis.
Abaqus
```
*STEP
*COMPLEX FREQUENCY, CORIOLIS=ON
20, ,
*DLOAD
element_set, CENTRIF, omega_squared, x, y, z
*END STEP
```
CORIOLIS=ON enables gyroscopic effect.
Ansys
```
/SOLU
ANTYPE, MODAL
MODOPT, QRDAMP, 20
CORIOLIS, ON, , ON ! Enable Coriolis/Gyroscopic effect
OMEGA, , , 100. ! z-axis rotation 100 rad/s
SOLVE
```
In Ansys, you enable the gyroscopic effect with the CORIOLIS command, I see.
Ansys's QRDAMP method projects onto real modes first, then solves for complex eigenvalues including damping/gyroscopic effects. It's efficient for large-scale models.
Creating a Campbell Diagram
Procedure:
1. Set rotational speed at 10-20 points (from 0 to maximum operating speed).
2. Perform prestressed modal + complex eigenvalue analysis at each speed.
3. Plot rotational speed vs. natural frequency.
4. Overlay excitation lines ($f = \Omega, 2\Omega, 3\Omega, ...$).
5. Identify intersections as critical speeds.
It's automated in Ansys Workbench, right?
Ansys Workbench's "Rotordynamics Analysis" can automatically generate Campbell Diagrams. It integrates parametric sweeps of rotational speed and result plotting.
Summary
Let me organize the numerical methods for rotating bodies.
Key Points:
- Complex Eigenvalue Analysis is Required — Complex eigenvalues due to gyroscopic matrix $[G]$.
- SOL 107 (Nastran), COMPLEX FREQUENCY CORIOLIS=ON (Abaqus), QRDAMP+CORIOLIS (Ansys)
- Automatically Generate Campbell Diagrams — Ansys is the most straightforward.
- Analyze Many Cases with Varying Rotational Speed — Parametric Study.
Creating Campbell Diagrams and Identifying Critical Speeds
A Campbell Diagram plots natural frequency on the vertical axis and rotational speed on the horizontal axis, showing natural frequency curves at each speed and straight lines of integer multiples of rotational speed (engine orders). Intersections are "critical speeds", and designs either avoid operation at those speeds or pass through resonance quickly. ISO 10816 requires evaluation of critical speeds using Campbell diagrams as a design standard for rotating machinery.
Practical CAE quality notes for Dynamic Analysis of Rotating Structures
Dynamic Analysis of Rotating Structures should be treated as an engineering model, not as an isolated formula. In structural analysis, reliable results come from a clear chain of assumptions: governing physics, material data, boundary conditions, numerical discretization, solver settings, and post-processing criteria. Before using this note in a design review, identify which quantities are prescribed, which are solved, and which are only diagnostic indicators.
Model setup checklist
- Define the scope: decide whether Dynamic Analysis of Rotating Structures is being used for screening, detailed design, failure investigation, or verification of another simulation.
- Check dimensions and units: keep SI units internally and document every conversion applied to loads, geometry, material constants, and time or frequency scales.
- State assumptions explicitly: record linearity, steady-state or transient behavior, small-deformation limits, continuum assumptions, and any symmetry or ideal boundary conditions.
- Compare with a baseline: use a hand calculation, limiting case, mesh refinement trend, or independent solver result before accepting the final value.
Validation signals
| Review item | What to verify | Typical warning sign |
|---|---|---|
| Inputs | Geometry, material data, loads, and constraints match the intended structural analysis problem. | Correct-looking plots with unrealistic magnitudes or units. |
| Numerics | Mesh, time step, convergence tolerance, and solver options are adequate for Rotating Structure. | Large changes after small mesh or tolerance adjustments. |
| Physics | The selected theory remains valid in the expected stress, temperature, velocity, or frequency range. | Results are used outside the assumptions stated in the model. |
For production use, keep the model file, input table, result plots, and review comments together. This makes Dynamic Analysis of Rotating Structures traceable and prevents the page from being used as a black-box answer without engineering judgment.
Related Topics
Experience the theory firsthand with the interactive simulator for this field
All Simulators