Complex Eigenvalue Analysis
Complex Eigenvalue Analysis: Theoretical Foundations
What are Complex Eigenvalues?
Professor, what's the difference between "complex eigenvalues" and regular eigenvalues?
Regular eigenvalue analysis for vibration (real eigenvalue analysis) deals with undamped free vibration. The eigenvalues are real numbers ($\omega^2$), yielding natural frequencies and mode shapes.
Complex eigenvalue analysis deals with damped free vibration. The eigenvalues become complex numbers ($\lambda = \sigma + i\omega_d$), where the real part $\sigma$ represents damping (stability), and the imaginary part $\omega_d$ represents the damped natural frequency.
Governing Equation
The eigenvalue problem with damping:
$\lambda$ and $\{\phi\}$ become complex numbers.
Complex eigenvalues... what's their physical meaning?
With $\lambda = \sigma + i\omega_d$:
- $\omega_d$ = Damped natural frequency (slightly lower than the real natural frequency $\omega_n$)
- $\sigma$ = Decay rate. $\sigma < 0$ indicates stability (vibration decays), $\sigma > 0$ indicates instability (vibration grows)
- $\zeta = -\sigma / \sqrt{\sigma^2 + \omega_d^2}$ = Damping ratio
When Complex Eigenvalues are Needed
| Situation | Reason |
|---|---|
| Brake squeal | Self-excited vibration due to friction. Detects unstable eigenvalues ($\sigma > 0$) |
| Flutter analysis | Aeroelastic instability. $\sigma$ becomes positive as speed increases |
| Non-proportional damping | When the damping matrix cannot be diagonalized by the modes |
| Rotating machinery | Gyroscopic effects make the damping matrix asymmetric |
So brake squeal is a typical application?
Friction between the brake pad and disc makes the stiffness matrix asymmetric. Asymmetric stiffness destabilizes the system, causing eigenvalues with $\sigma > 0$ (self-excited vibration modes) to appear. These modes cause brake squeal. It's one of the most important analyses in automotive NVH development.
Nastran
```
SOL 107 $ Complex eigenvalue analysis
CEND
CMETHOD = 10
BEGIN BULK
EIGC, 10, HESS, , , , , 20
```
Abaqus
```
*STEP
*COMPLEX FREQUENCY
20, ,
*END STEP
```
SOL 107 for Nastran, COMPLEX FREQUENCY for Abaqus.
The setup is similar to regular eigenvalue analysis, but defining the damping matrix $[C]$ is essential. Without damping, you get the same result as real eigenvalue analysis.
Summary
Let me summarize complex eigenvalue analysis.
Key points:
- Eigenvalue problem including damping — Eigenvalues are complex numbers $\lambda = \sigma + i\omega_d$
- $\sigma > 0$ indicates instability (self-excited vibration) — Detection of brake squeal, flutter
- Handles non-proportional damping and asymmetric stiffness — Cannot be handled by real eigenvalue analysis
- SOL 107 (Nastran), *COMPLEX FREQUENCY (Abaqus)
- Main applications are brake NVH and aerospace flutter
So if real eigenvalue analysis is for "knowing the natural frequencies of a structure," complex eigenvalue analysis is for "checking if the structure becomes unstable."
Exactly. Complex eigenvalue analysis is a stability analysis and an extension of real eigenvalue analysis.
Unstable Mechanism of Brake Squeal
Disk brake "squeal" is caused by instability in complex eigenvalues (positive real part). Friction increases the vibrational energy of the structure, and when the real part becomes positive, the vibration diverges. This phenomenon is essentially the same mechanism as Flutter instability, and know-how from aerodynamic Flutter research (1940s) was applied to brake analysis in the 1990s.
Computational Methods for Complex Eigenvalue Analysis
Complex Eigenvalue Solvers
How are complex eigenvalues solved?
Different algorithms are needed compared to the Lanczos method for real eigenvalues.
| Method | Characteristics | Solver |
|---|---|---|
| Hessenberg method | Small to medium scale. Finds all eigenvalues | Nastran EIGC(HESS) |
| QZ method | Generalized eigenvalue problem. Stable | LAPACK |
| Projection method | Projects onto real modes, then complexifies | Abaqus COMPLEX FREQUENCY |
| Arnoldi method | Large-scale sparse matrices. Asymmetric version of Lanczos | For research |
What is Abaqus's "projection method"?
First, obtain $N$ modes from real eigenvalue analysis, then solve a small complex eigenvalue problem projected onto the real mode space. The complex eigenvalues of the small $N \times N$ matrix can be solved by the QZ method. Can handle large-scale problems.
So the real modes become the "basis," right?
Therefore, before complex eigenvalue analysis, you must always obtain a sufficient number of real modes. Insufficient mode count reduces the accuracy of complex eigenvalues.
Brake Squeal Analysis Setup
Procedure for complex eigenvalue analysis of brake squeal:
1. Nonlinear static analysis including frictional contact — Determine the clamped state of the brake
2. Friction force on contact surface → Linearization — Construct asymmetric stiffness matrix from friction force
3. Complex eigenvalue analysis — Search for unstable eigenvalues ($\sigma > 0$)
4. Identify unstable modes — Squeal frequency and mode shape
So the asymmetric stiffness from friction is the cause of instability.
Friction force is a follower force (a force whose direction changes following displacement), making $[K]$ asymmetric. Asymmetric $[K]$ can "inject energy" into the system, causing unstable modes with $\sigma > 0$ to appear.
Summary
Let me summarize the numerical methods for complex eigenvalues.
Key points:
- Projection method (real modes → complex eigenvalues) is standard in Abaqus — Real mode count dictates accuracy
- Hessenberg/QZ methods are standard in Nastran — Small to medium scale
- Brake squeal — Asymmetric stiffness from friction → detects unstable modes via complex eigenvalues
- Modes with $\sigma > 0$ are unstable (self-excited vibration) — Eliminate via design changes
Why Asymmetric Stiffness Matters
In classical vibration theory, symmetric matrices ensure stable, passive behavior. Friction forces and follower forces introduce asymmetric terms into the stiffness matrix, which can extract energy from steady-state conditions and amplify oscillations. This is the physical root of brake squeal and flutter phenomena.