Dynamic Analysis of Rotating Structures
Theory and Physics
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].
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Haven't you experienced being thrown forward when braking suddenly? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, which is the assumption that "acceleration can be ignored because forces are applied slowly". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ and $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it", right? That is Hooke's law $F=kx$, the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more when pulled with the same force? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness = strong" is not correct. Stiffness is "resistance to deformation", strength is "resistance to failure"; they are different concepts.
- External Force Term (Load Term): Body force $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire contents" (body force), the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression" — sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades away. That's because vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — deliberately absorbing vibration energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't actually happen, setting appropriate damping is important.
Assumptions and Applicability Limits
- Continuum Assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small Deformation Assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and Stress-strain relationship is linear.
- Isotropic Material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-Static Assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-Applicable Cases: Large Deformation/Large Rotation problems require geometric nonlinearity. Nonlinear material behavior like Plasticity or creep requires constitutive law extensions.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistencies when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation). |
| Elastic Modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system. |
Numerical Methods and Implementation
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.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Computational cost is low, but stress accuracy is low. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with Mid-Side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommendation: When stress evaluation is important.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (Locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
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