Frequency Response of Viscoelastic Damping Materials
Frequency Response of Viscoelastic Damping Materials: Theoretical Foundations
What is a Viscoelastic Material?
Professor, how is the damping of viscoelastic materials different from structural damping?
Structural damping assumes $g$ is independent of frequency, but the damping of viscoelastic materials strongly depends on frequency and temperature. Rubber, polymers, adhesives, and damping materials are typical examples.
Complex Modulus
Stress-strain relationship for viscoelastic materials:
- $E'(\omega)$ — Storage modulus (stiffness. Stores energy)
- $E''(\omega)$ — Loss modulus (damping. Dissipates energy)
- $\eta(\omega) = E''/E'$ — Loss factor
So both stiffness and damping change with frequency, right?
Typical behavior:
- Low frequency — $E'$ is low (soft), $\eta$ is moderate
- Glass transition region — $E'$ increases sharply, $\eta$ peaks (maximum damping)
- High frequency — $E'$ is high (stiff), $\eta$ decreases
Temperature-Frequency Equivalence Principle (WLF Equation)
For viscoelastic materials, when temperature increases, the properties shift to the low-frequency side, and when temperature decreases, they shift to the high-frequency side. This equivalence is described by the Williams-Landel-Ferry (WLF) equation:
$a_T$ is the shift factor. This allows estimating properties at all temperatures from data at a single temperature.
So if we measure data at one temperature using DMA (Dynamic Mechanical Analysis), we can extrapolate to other temperatures using the WLF equation, right?
Create a master curve. Shift the DMA data at the reference temperature using the WLF equation to obtain $E'(\omega), \eta(\omega)$ over a wide frequency range. Input this into FEM.
Summary
Key points:
- $E^*(\omega) = E'(1+i\eta)$ — Frequency-dependent complex modulus
- Both $E'$ and $\eta$ depend on frequency and temperature — Peak at glass transition
- Temperature-frequency equivalence via WLF equation — Construction of master curve
- Input into FEM using direct method (frequency response) — Difficult to handle with modal methods
The memory effect of viscoelasticity was discovered by Maxwell in 1867
The memory effect of viscoelastic materials, where "strain depends on the history of stress," was discovered by James Clerk Maxwell in 1867 during his research on gas viscosity. Later, Kelvin-Voigt (1890) proposed a parallel model, and the "Zener (standard linear solid) model," which combines the Maxwell series and Kelvin parallel models, became the foundation for FEM implementation of structural damping. This model is still used today for characterizing the properties of 3M's damping sheets.
Computational Methods for Frequency Response of Viscoelastic Damping Materials
Viscoelastic Materials in FEM
How are viscoelastic materials handled in FEM?
Model $E'(\omega), E''(\omega)$ using the Prony series:
$E_i, \tau_i$ are Prony series parameters. $N = 5 \sim 15$ terms cover a wide frequency range.
Abaqus
```
*VISCOELASTIC, FREQUENCY=PRONY
g_1, k_1, tau_1
g_2, k_2, tau_2
...
```
Nastran
```
MAT1, 1, ...
TABLEM1, 100, ...
$ Define frequency-dependent E' and η in a table
```
In Nastran, you can input directly via a table (frequency vs. E', η).
Can Abaqus's Prony series be fitted directly from DMA data?
Abaqus/CAE's *VISCOELASTIC, TEST DATA option allows input of measured $E'(\omega), E''(\omega)$ and automatically fits the Prony series. Very convenient.
Damping Material (CLD) Analysis
Analysis of CLD (Constrained Layer Damping):
1. Model the laminate: substrate (steel plate, etc.) + viscoelastic layer + constraining layer
2. Set frequency-dependent Prony series for the viscoelastic layer
3. Evaluate damping effect using direct method frequency response analysis
4. Confirm reduction of FRF peaks
Summary
Compress data to 1/100 using time-temperature superposition
The loss factor of viscoelastic materials changes with a combination of temperature and frequency. Using the time-temperature superposition principle via the Williams-Landel-Ferry (WLF) equation (1955), properties at any temperature and frequency can be predicted from measurement data at a reference temperature Tr alone. The frequency characteristics of interior materials (ethylene-propylene rubber based) in vehicles can be represented by a single master curve at 20°C reference for all conditions from -40°C to 100°C and 1Hz to 10kHz.
Frequency Response of Viscoelastic Damping Materials in Practice
Practical Aspects of Viscoelastic Damping
Damping material design is one of the most important applications in automotive NVH.
Design Flow of Vibration Damping Materials
1. Measure $E'(\omega, T), \eta(\omega, T)$ of the viscoelastic material via DMA testing
2. Construct master curve using WLF equation
3. Fit to Prony series
4. Optimize damping material placement and thickness in FEM
5. Confirm reduction of FRF resonance peaks
Practical Checklist
So damping material should be applied at the "location of maximum strain"?
Damping materials dissipate energy through shear deformation. Applying damping material at the location of maximum bending strain in the substrate (near fixed ends) maximizes effectiveness. Applying it at locations with low strain (near free ends) has little effect.
3M damping sheet reduces Shinkansen noise by 5dB
3M's damping material series "Dynamat" is a structure with a viscoelastic polymer (acrylic-based) laminated onto a steel sheet, converting bending waves at 100-1000Hz into thermal energy. An equivalent product to 3M Dynamat was applied to the underfloor steel plates of JR East's E5 series Shinkansen (2011), reducing interior structure-borne noise from track noise by approximately 5dB. It is only 2mm thick with an area density of 2.4kg/m². Due to cost advantages, it is widely adopted in mass-produced vehicles.