Identification and Setting of Modal Damping
Theory and Physics
What is Damping?
Professor, how is "damping" handled in structural mechanics?
Damping is the effect of dissipating vibrational energy. Without damping, a structure would vibrate forever. Real structures always have damping, which keeps the amplitude finite during resonance.
Equation of Motion
The equation of motion with damping:
$[C]$ is the damping matrix. $[M]$ and $[K]$ are physically clear, but $[C]$ generally has large uncertainty.
So the problem is how to determine the damping matrix.
Exactly. Damping modeling is the most uncertain parameter in FEM dynamic analysis.
Damping Models
Major damping models:
1. Modal Damping
Directly specify a damping ratio $\zeta_i$ for each mode. The most common method.
$q_i$ is the modal coordinate, $\omega_i$ is the natural angular frequency.
2. Rayleigh Damping
$[C] = \alpha [M] + \beta [K]$. $\alpha$ and $\beta$ are determined by matching $\zeta$ at two frequencies.
3. Structural Damping
Also called hysteretic damping. Frequency-independent damping, expressed with complex stiffness $[K^*] = K$. $g$ is the structural damping coefficient.
How do you choose which of the three models to use?
| Model | Application | Advantage | Disadvantage |
|---|---|---|---|
| Modal Damping | Mode superposition method | Different $\zeta$ for each mode | Requires modal analysis |
| Rayleigh Damping | Direct integration method (time history analysis) | Usable in time domain | Can only match $\zeta$ at two frequencies |
| Structural Damping | Frequency Response Analysis | No frequency dependence | Cannot be used in time domain |
Typical Damping Ratio Values
| Structure | Damping Ratio $\zeta$ |
|---|---|
| Steel Structure (welded) | 0.5〜1% |
| Steel Structure (bolted) | 1〜2% |
| RC Structure | 3〜5% |
| Seismic Isolation Structure | 10〜30% |
| Mechanical Structure | 1〜3% |
| Composite Structure | 0.5〜2% |
0.5〜1% for steel structures... that's very small.
Steel has low internal damping. That's why steel structures are prone to large amplitudes at resonance, and damping settings greatly affect the results.
Summary
Let me organize the theory of modal damping.
Key points:
- Damping is the most uncertain parameter in dynamic analysis — sensitivity analysis is essential
- Three models — Modal damping, Rayleigh damping, Structural damping
- Modal damping is the most common — specify $\zeta_i$ for each mode
- Rayleigh damping is for time domain — match two frequencies with $\alpha, \beta$
- Typical damping ratio values — Steel: 1%, RC: 5%. Varies by application.
So the damping setting can change the results by many times. Since the amplitude at resonance is proportional to $1/(2\zeta)$, the amplitude differs by a factor of 2 between $\zeta = 1\%$ and $\zeta = 2\%$.
That's why damping is "the parameter with the greatest impact and the greatest uncertainty". You should not trust the results of a dynamic analysis without sensitivity analysis for damping.
The "Magic Number" of 2% Damping Ratio
The structural damping ratio ζ=2% is widely used as a design convention, but actual steel structures vary significantly from 0.5% to 5%. This value was statistically proposed by Lankford (1954) from measured data of buildings. It became established as a "standard value" after being adopted in the 1970s UBC (Uniform Building Code), but we must not forget that welded structures are less than 1%, and bolted structures are 3-5%, which are completely different.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Haven't you experienced being thrown forward when slamming on the brakes? 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 assumes "forces are applied slowly so acceleration can be ignored". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it", right? That's 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". 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 rotate 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. That's because vibrational energy turns into heat due to air resistance and internal friction in the string. Car shock absorbers work on the same principle — intentionally absorbing vibrational energy to improve ride comfort. What if damping were zero? Buildings would keep shaking forever after an earthquake. Since that doesn't actually happen, appropriate damping settings are 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 equilibrium between external and internal forces.
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Plasticity, creep, and other nonlinear material behaviors require 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 inconsistency 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
How to Set Damping
How do you set damping in FEM?
Modal Damping in Nastran
```
TABDMP1, 1, CRIT
, 0., 0.02, 100., 0.02, ENDT
```
Sets $\zeta = 2\%$ for all modes. Different $\zeta$ for different frequency ranges is also possible.
Modal Damping in Abaqus
```
*MODAL DAMPING
1, 50, 0.02
```
Sets $\zeta = 2\%$ collectively for modes 1 to 50.
Rayleigh Damping Settings
How to determine $\alpha$ and $\beta$:
When setting $\zeta_1 = \zeta_2 = \zeta$ at two frequencies $f_1, f_2$:
How do you choose $f_1$ and $f_2$?
The lower and upper limits of the frequency range of interest. For example, for seismic response targeting 1-10 Hz, use $f_1 = 1$ Hz, $f_2 = 10$ Hz. Note that the damping ratio will deviate outside this range.
Rayleigh Damping in Abaqus
```
*DAMPING, ALPHA=0.5, BETA=0.001
```
Damping Identification
How do you measure the damping ratio of a real structure?
Measure with Experimental Modal Analysis:
1. Hammer Excitation Method — Excite with an impulse hammer and measure acceleration.
2. Shaker Method — Obtain Frequency Response Function (FRF) with sine/random excitation.
3. Damping Identification — Half-power bandwidth method or curve fitting from FRF.
Half-power bandwidth method: From two frequencies $f_1, f_2$ where the amplitude becomes $1/\sqrt{2}$ of the resonance peak:
So it can be measured easily.
The principle is simple, but experimental accuracy (excitation point, measurement point, noise processing) affects the results. It's safer to cross-validate with multiple methods.
Summary
Let me organize the numerical methods for damping settings.
Key points:
- TABDMP1 (Nastran), *MODAL DAMPING (Abaqus) — Setting modal damping.
- Determining $\alpha, \beta$ — Matching at two frequencies.
- Measure damping ratio with Experimental Modal Analysis — Half-power bandwidth method is basic.
- Be careful of the damping ratio range — Rayleigh damping deviates outside the specified range.
- If no experimental data, use literature values and perform sensitivity analysis.
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