Damping — CAE Terminology Glossary
What is Damping
What exactly is damping? Is it just when vibrations gradually get smaller?
Yes, it's the phenomenon where the energy of a vibrating system is dissipated as heat or sound, causing the amplitude to gradually decrease. A car suspension shock absorber is a typical example. After going over a bump, vibrations quickly die down thanks to damping.
What happens if there's no damping?
With zero damping, once vibration starts, it never stops. Moreover, if resonance occurs, theoretically the amplitude becomes infinite. In reality, material yielding or failure stops it, but the 1940 Tacoma Narrows Bridge collapse is a famous case where insufficient damping and resonance combined with catastrophic results.
Viscous Damping and Structural Damping
What's the difference between viscous damping and structural damping? I'm confused because both appear in my textbook.
Viscous Damping is a damping force proportional to velocity, like a dashpot. In the equation of motion it's written as
where $c$ is the damping coefficient with units N·s/m. This is like hydraulic dampers that dissipate energy through fluid viscosity.
So what's the mechanism behind structural damping?
Structural Damping (Hysteretic Damping) models energy dissipation from microscopic plastic deformation and grain boundary sliding within materials. The damping force is proportional to displacement but with a 90-degree phase shift, expressed using complex stiffness:
Here $\eta$ is called the Loss Factor. A key feature of structural damping is that it's frequency-independent. For steel structures, $\eta$ is typically 0.001 to 0.01; for rubber, it can reach 0.1 to 1.0. In reality, most structures behave more like structural damping than viscous damping.
Which one should I use?
For frequency-domain analysis (harmonic response analysis and others), structural damping is naturally handled. For time-domain analysis (transient response analysis and others), viscous damping is numerically more convenient. In practice, people often convert between them so that the damping ratio is equivalent in the frequency band of interest.
Critical Damping and Damping Ratio
What is critical damping? I see "damping ratio $\zeta = 1$" written but I don't quite grasp it.
For a single-degree-of-freedom system with equation of motion
the critical damping is when the discriminant of the characteristic equation equals zero. The critical damping coefficient is
$$c_{cr} = 2\sqrt{km} = 2m\omega_n$$and the damping ratio is defined as this ratio:
$$\zeta = \frac{c}{c_{cr}} = \frac{c}{2m\omega_n}$$How does vibration behavior change with different values of $\zeta$?
There are three cases:
- $\zeta < 1$ (Underdamping): Oscillates while damping. Most engineering structures fall in this category. For steel structures, $\zeta \approx 0.01$ to $0.02$; for concrete, $0.03$ to $0.07$ is typical
- $\zeta = 1$ (Critical Damping): Returns to equilibrium most quickly without oscillation. Door closers are designed near this condition
- $\zeta > 1$ (Overdamping): No oscillation but slower return. Used in precision instruments where overshoot must be avoided
The free vibration solution for underdamped systems is
$$x(t) = X e^{-\zeta\omega_n t} \cos(\omega_d t - \phi)$$where $\omega_d = \omega_n\sqrt{1-\zeta^2}$ is the damped natural frequency. For small $\zeta$, $\omega_d \approx \omega_n$, but when $\zeta$ exceeds 0.3, the difference becomes significant.
Rayleigh Damping
I often see "Rayleigh damping" in CAE dynamic analysis. What kind of damping model is this?
The equation of motion for a multi-degree-of-freedom system is
The challenge is how to construct the damping matrix $[C]$. Rayleigh damping constructs it as
$$[C] = \alpha [M] + \beta [K]$$a linear combination of the mass and stiffness matrices. It's defined by two parameters: $\alpha$ (mass-proportional term) and $\beta$ (stiffness-proportional term).
How do you determine $\alpha$ and $\beta$?
The damping ratio for the $i$-th mode is
So if you specify target damping ratios $\zeta_1$ and $\zeta_2$ at two angular frequencies $\omega_1$ and $\omega_2$, you can solve for $\alpha$ and $\beta$ from simultaneous equations. A common practice is to assume $\zeta_1 = \zeta_2 = \zeta$ (same damping ratio everywhere), which gives
$$\alpha = \frac{2\omega_1 \omega_2}{\omega_1 + \omega_2}\zeta, \quad \beta = \frac{2}{\omega_1 + \omega_2}\zeta$$Any tips for choosing $\omega_1$ and $\omega_2$?
This is the key point with Rayleigh damping. Between $\omega_1$ and $\omega_2$, the damping ratio is less than the target (underdamping), and outside this range it's greater (overdamping). So always choose $\omega_1$ and $\omega_2$ to cover the dominant frequency band contributing to the response. For seismic analysis, for example, you'd select the first natural frequency and frequencies that encompass the dominant earthquake spectrum. Many practitioners set $\omega_1$ at 0.5 to 1 times the first natural frequency and $\omega_2$ around the 3rd to 5th natural frequency.
Modal Damping Ratio
I heard you can assign damping ratios directly to each mode in modal analysis. How is this different from Rayleigh damping?
Modal Damping is a method used with the mode superposition method where you assign individual damping ratios $\zeta_i$ directly to each mode. Unlike Rayleigh damping, it's not constrained by the U-shaped frequency dependence, so you can set precise damping ratios for each mode based on experimental data.
Wouldn't it be better to always use modal damping then?
Modal damping only works with the mode superposition method. Explicit dynamic and implicit dynamic direct time integration methods need a damping matrix $[C]$, so you must use Rayleigh damping. Also, in nonlinear analysis, the mode shapes change, so mode superposition can't be used. That's why the basic rule is: use modal damping for linear mode methods, and Rayleigh damping for direct integration or nonlinear analysis.
How do you determine modal damping ratios from experiments?
There are three main methods:
- Half-Power Bandwidth Method: Uses the bandwidth $\Delta\omega$ where the frequency response function drops 3 dB from the peak: $\zeta = \Delta\omega / (2\omega_n)$
- Logarithmic Decrement: From successive peak ratios in free vibration: $\delta = \ln(x_n / x_{n+1})$, then $\zeta = \delta / \sqrt{4\pi^2 + \delta^2}$
- Curve Fitting: Fit the frequency response function to theoretical equations. Effective when multiple modes are close together
For automotive bodies, damping ratios vary by mode from 0.01 to 0.05, so experimental identification is worthwhile instead of using a single uniform value.
Damping Settings in CAE Practice
When setting damping in CAE without experimental data, how do you proceed?
Using literature values and standard recommendations is practical. Here are some typical values:
- Steel structure (welded): $\zeta = 0.02$ to $0.03$
- Steel structure (bolted): $\zeta = 0.03$ to $0.05$ (increases due to joint friction)
- Reinforced concrete: $\zeta = 0.03$ to $0.07$
- Aluminum alloy: $\zeta = 0.002$ to $0.01$
- Rubber and polymers: $\zeta = 0.05$ to $0.15$
However, slippage at bolted joints and effects of non-structural components often result in higher measured damping. When in doubt, run parametric studies with multiple damping ratios and check result sensitivity—that's good practice.
What happens if you set damping too high?
Response gets underestimated, leading to unconservative design. Especially with Rayleigh damping, if you set $\omega_2$ too low, higher modes get excessive damping, which incorrectly suppresses high-frequency response components like those from impact loads. Conversely, setting damping too low overestimates response—conservative but uneconomical. Either way, damping is the parameter with the largest uncertainty and biggest effect on analysis results. That's what you need to remember.
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