Structural Damping (Hysteretic Damping) — Complex Stiffness, Loss Factor, FEM Implementation
Theory and Physics
Definition of Complex Stiffness and Loss Factor
Professor, the term "structural damping" came up in vibration analysis. How is it different from regular damping? It's different from something like a damper, right?
Good question. The damping force of a dashpot (viscous damper) is proportional to velocity — $F_d = c\dot{x}$. This is viscous damping. On the other hand, structural damping (hysteretic damping) represents energy dissipation due to microscopic friction and slip within the material. Mathematically, it is expressed using complex stiffness:
Here, $\eta$ is the loss factor. The real part $K$ represents the usual elastic stiffness, and the imaginary part $K\eta$ represents the magnitude of energy dissipation.
What does it mean physically to represent stiffness with a complex number? It's hard to imagine an imaginary spring...
It helps to think of it this way. Assuming harmonic vibration $x = X e^{i\omega t}$, the restoring force is:
The real part is the elastic restoring force in phase with displacement, and the imaginary part is a force 90° ahead of displacement — corresponding to a dissipative force related to velocity. Comparing this to the viscous damping force $c\dot{x} = ci\omega x$, the dissipative force in structural damping is independent of $\omega$. This is the crucial difference.
Fundamental Difference from Viscous Damping
What kind of practical difference does the frequency independence make? For example, in automotive vibration analysis.
In viscous damping, the equivalent damping ratio is proportional to frequency:
So the damping ratio differs by a factor of 10 between 20 Hz and 200 Hz. When dealing with a broad frequency band from 50 to 500 Hz in automotive dashboard vibration analysis, using only viscous damping results in insufficient damping at low frequencies and excessive damping at high frequencies.
With structural damping, the equivalent damping ratio is:
Constant across all frequencies. The actual damping characteristics of metal or FRP structures are much closer to the structural damping model. That's why structural damping is standard in frequency response analysis (NVH analysis).
So the relationship between $\eta$ and $\zeta$ is $\eta \approx 2\zeta$, right? Can we convert if we know the damping ratio from modal analysis?
Exactly. As an equivalent relationship at the resonance frequency:
For example, if experimental modal analysis identifies $\zeta = 1\%$, then set $\eta = 0.02$. However, note that this equivalence holds strictly only at the resonance frequency. The response of viscous and structural damping differs at frequencies away from resonance.
Energy Dissipation and Hysteresis Loop
Is it called "hysteretic damping" because it's related to the hysteresis loop in the stress-strain curve?
Precisely. When cyclic loading is applied to a material, the stress-strain curve forms a closed loop. The area of this loop corresponds to the energy dissipation per cycle $\Delta W$. The loss factor can be defined as the ratio of dissipated energy per cycle to the maximum elastic energy:
Here, $W_{\max} = \frac{1}{2}K X^2$ is the maximum elastic strain energy. For example, the loop area for steel is very small, so $\eta \approx 0.001$〜$0.01$, while the loop for rubber is large, so $\eta \approx 0.2$〜$1.0$. Literally, the "size of the hysteresis" determines the damping strength.
Identification of Loss Factor via Half-Power Bandwidth Method
How can we measure the loss factor experimentally? Can it be determined from an impulse hammer test?
The most common method is the Half-Power Bandwidth Method. It reads the loss factor from the resonance peak of the FRF. The procedure is as follows:
- Identify the resonance frequency $f_n$ and the maximum amplitude $|H|_{\max}$ of the FRF peak.
- Find the two frequencies $f_1, f_2$ where the amplitude is $|H|_{\max}/\sqrt{2}$ (= -3 dB).
- Calculate the loss factor:
For example, if an impulse hammer test shows a resonance at 200 Hz and a -3 dB bandwidth of 2 Hz, then $\eta = 2/200 = 0.01$. This corresponds to $\zeta = 0.5\%$, a typical value for welded steel structures.
The half-power bandwidth method doesn't seem to work well for closely spaced modes, does it?
Sharp observation. When modes are dense, the -3 dB line overlaps with adjacent peaks, making accurate reading difficult. In such cases, use the Circle Fit method (fitting an arc on the Nyquist plot) or Curve Fitting methods (like RFP, LSCE). Especially for materials like CFRP laminates with high damping and overlapping modes, it's better to avoid the simple half-power bandwidth method.
Complex Stiffness Originated from Aircraft Flutter Analysis
The model representing structural damping as complex stiffness $K^* = K(1+i\eta)$ was born in the 1930s from aircraft wing flutter research. To solve the coupled vibration of elastic wing deformation and aerodynamic forces, a concise way to express the material's dissipative properties was needed. In 1960, T.K. Caughey of Caltech rigorously organized the equivalence between complex stiffness and viscous damping, establishing the practical formula $\zeta = \eta/2$. It is still used today as the GE parameter in Nastran.
Numerical Methods and Implementation
Structural Damping in Direct vs. Modal Methods
Structural damping is used in frequency response analysis, right? Do we use it in the direct method or the modal method?
It can be used in both. In the direct method (Nastran SOL 108), complex stiffness is directly substituted into the equation of motion:
In the modal method (Nastran SOL 111), damping is applied per mode after expanding into eigenmodes:
Here, $q_r$ is the modal coordinate and $\phi_r$ is the mode vector. In practice, the modal method is overwhelmingly more common. The computational cost is 1/10 to 1/100 of the direct method.
In the modal method, can we vary $\eta_r$ for each mode? It's common for experimental damping ratios to differ per mode, right?
Using Nastran's TABDMP1 card, you can define a table of damping versus frequency. It automatically interpolates $\eta$ based on each mode's natural frequency. For example, if experimental modal analysis identifies the 1st mode $\zeta_1=0.5\%$ and the 2nd mode $\zeta_2=1.2\%$, then enter $\eta_1=0.01$, $\eta_2=0.024$ into the table.
Frequency Response Function (FRF) Formulation
Please show me the FRF formula including structural damping. I want to know how the amplitude at resonance changes.
The FRF (receptance) for a 1-DOF system with structural damping is:
Here, $r = \omega/\omega_n$ (frequency ratio). At the resonance point $r=1$:
This means the resonance amplitude is $1/(K\eta)$, inversely proportional to the loss factor. If you get the order of magnitude of $\eta$ wrong by one digit, the resonance amplitude becomes 10 times larger, so always double-check the input value. There was an actual case in automotive steering column vibration analysis where $\eta=0.02$ was mistakenly input as $\eta=0.002$, causing the FRF peak to be 10 times larger and causing a big stir.
Conversion Technique to Rayleigh Damping
Structural damping can't be used in time history response analysis, right? When converting to Rayleigh damping, how do we determine $\alpha$ and $\beta$?
Rayleigh damping is $C = \alpha M + \beta K$, and the damping ratio is:
Since structural damping has a constant $\zeta = \eta/2$, we solve simultaneous equations to satisfy $\zeta(\omega_i) = \eta/2$ at two frequencies $\omega_1, \omega_2$:
For example, with $\eta = 0.02$ and a frequency band of interest from 50 to 300 Hz ($\omega_1 = 100\pi, \omega_2 = 600\pi$), calculate $\alpha, \beta$ so that $\zeta = 1\%$ at these two points. Note that damping will be underestimated between these two points and overestimated outside them. The results can vary significantly depending on the chosen band, so be careful.
Causality Constraint and Handling in the Time Domain
Could you explain in more detail why structural damping can't be used in the time domain? What does "violation of causality" mean specifically?
If you inverse Fourier transform the complex stiffness $K(1+i\eta)$ back to the time domain, the impulse response function $h(t)$ does not become zero for $t < 0$. This means a response appears before the input arrives — violating causality (the time order of cause → effect).
This is because the assumption that the loss factor $\eta$ is constant across all frequencies itself violates the Kramers-Kronig relations (constraints between real and imaginary parts required to satisfy causality). A physically rigorous damping model must have frequency dependence. Structural damping should be used as a "frequency-domain approximation model" with that understanding.
The Half-Power Bandwidth Method Has Been Used for Over 60 Years
The Half-Power Bandwidth Method calculates the loss factor $\eta = \Delta f / f_n \approx 2\zeta$ from the two frequencies at the $1/\sqrt{2}$ amplitude of the FRF peak. Since D.J. Ewins systematized it in the 1960s, it remains a standard technique in vibration testing. The $\eta$ values vary by over two orders of magnitude depending on the material: steel ~0.001–0.01, CFRP composites ~0.005–0.02, and damping steel sheets ~0.05–0.2.
Practical Guide
Typical Loss Factors by Material/Structure
What are typical values for the loss factor $\eta$? What should I reference when inputting it for analysis?
Let me summarize some representative values. First, loss factors for materials alone:
| Material | Loss Factor $\eta$ | Equivalent Damping Ratio $\zeta$ | Notes |
|---|---|---|---|
| Pure Aluminum | 0.0001–0.001 | 0.005–0.05% | Depends on purity/crystal structure |
| Aluminum Alloy (A6061, etc.) | 0.001–0.005 | 0.05–0.25% | Common in aircraft structures |
| Mild Steel (SS400, etc.) | 0.001–0.006 | 0.05–0.3% | Material-only value |
| Steel Structure (Welded) | 0.005–0.015 | 0.25–0.75% | Includes friction at joints |
| Steel Structure (Bolted) | 0.01–0.05 | 0.5–2.5% | Micro-slip at bolt interfaces dominates |
| CFRP Laminate | 0.005–0.02 | 0.25–1% | Depends on layup/matrix |
| Concrete | 0.02–0.1 | 1–5% | Varies greatly with crack state |
| Rubber (Natural Rubber) | 0.1–0.5 | 5–25% | Strongly dependent on temperature/frequency |
| Damping Rubber / Butyl Rubber | 0.3–1.5 | 15–75% | Used as damping material |
| Damping Steel Sheet (Sandwich) | 0.05–0.3 | 2.5–15% | Constrained layer damping material |
Bolted structures have higher damping than the material alone, huh? Why is that?
Good observation. According to Beards (1983), 60–90% of the total damping in mechanical structures originates from friction at joints. In bolted interfaces, micro-slip occurs during vibration, dissipating energy through Coulomb friction. That's why, even with the same steel, bolted structures ($\eta \approx 0.03$) have nearly three times the damping of welded structures ($\eta \approx 0.01$). When FEM results don't match measurements after inputting only material damping, this is usually the reason.
Application Example in Automotive NVH Analysis
In automotive NVH analysis using structural damping, how is it specifically set up?
Frequency response analysis of an automotive body (BIW: Body In White) is a classic application of structural damping. Typical settings are summarized as follows:
- Steel Body Panels: $\eta = 0.01$–$0.02$ (includes friction at spot welds)
- Dashboard with Damping Material Applied: $\eta = 0.02$–$0.05$ (effect of constrained layer damping material)
- Rubber Mounts (Engine/Subframe): $\eta = 0.1$–$0.3$
- Glass: $\eta = 0.002$–$0.005$
The standard approach is to perform a modal method analysis with Nastran SOL 111, setting $\eta$ per material using the GE field in MAT1. Applying a uniform value globally via PARAM, G is a rough method; for accuracy, material-specific settings are essential.
Application in Aerospace and Plant Piping
What about in aerospace? Is it different from automotive?
In aerospace, because damping is smaller, the setting values are even more critical. Some concrete examples:
- Satellite Panels (CFRP/Aluminum Honeycomb): $\eta = 0.005$–$0.01$. Used in launcher vibration environment analysis. Overestimating $\eta$ leads to underestimating peak stress, risking failure in orbit.
- Jet Engine Blades: $\eta = 0.001$–$0.003$. Accurate prediction of resonance amplitude is essential for high-cycle fatigue evaluation.
- Plant Piping Systems: $\eta = 0.01$–$0.03$ (includes friction at supports/clamps). ASME/JSME standards recommend a design damping ratio of $\zeta = 1$–$2\%$ ($\eta = 0.02$–$0.04$) for piping systems.
Practical Checklist
Here's a summary of items to check when setting structural damping:
- Confirm Analysis Type: It must be a frequency response analysis (SOL 108/111, *STEADY STATE DYNAMICS, HROPT). Cannot be used in time history analysis.
- Loss Factor Order of Magnitude: If $\eta > 0.1$ for a metal structure, suspect an input error.
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