Damping ratio

Hi,
seems like i'll be the first one to post a question.
I'm not familiar with how the damping ratio is calculated from the phase angle. A quick a dirty explanation or a ref where i can read about it would be appreciated.
thanks

Alan

Damping ratio Calculation
Dynamic complex modulus E* has a real and imaginary part that defines the elastic and viscoelastic behavior of the linear viscoelastic materials. It can be expressed as,

where E* is dynamic complex modulus and ζ is damping ratio. The E* can also be expressed in the form of,

where E' and E" are real (storage modulus) and imaginary (loss modulus) part of complex modulus .

Using equations 1 and 2, E’ and E” can be expressed in terms of ζ or ϕ (phase angle)

From equations 3 and 4, the damping ratio can be written as,

Please refer to “Pavement Strain from moving dynamic 3D load distribution” (R. V. Siddharthan at al., J. Trans. Eng., Vol. 124, No. 6 Nov/Dec 1998) paper for detailed explanation.

As described in the equation 5, damping ratio is a function of phase angle. Therefore it is important to understand the construction of phase angle master curve. Generally, there are four ways to develop the phase angle master curve and are briefly described below (K. P. Biligiri et al. Int. J. of Pav. Eng., Vol. 11, No. 2, pp. 143-152, April 2010):

1. Normal distribution – Phase angle master curve
This method uses the following equation to construct the phase angle master curve mathematically.

2. Beta Distribution - Phase angle Master Curve
Beta distribution method uses the following equation to develop the phase angle master curve mathematically. Microsoft excel solver or any other optimization tools are required to optimize the variable listed in this equation.


3. Sigmoidal function of dynamic, storage and loss moduli
The current version of 3D-Move uses this approach to develop the phase angle master curve. Initially |E*| master curve is developed mathematically using the following equation:

Similarly E' (storage modulus) master curve is constructed in the same manner. Then loss modulus E" master curve is constructed by using following equation.

Then phase angle is predicted by using the following equation.


4. Phase angle predicted Equation
Dynamic modulus laboratory data is required to develop the phase angle master curve for the first three methods. In case where dynamic modulus lab data is not available, this equation can be used to predict the phase angle from binder and mixture properties.


Though, presently 3D-Move uses sigmoidal method (i.e. Method 3) to predict the phase angle master curve, work is undergoing to incorporate the beta distribution method (i.e. Method 2) and the predicted equation method (i.e. Method 4) to develop phase angle master curve in 3D-Move. Consequently, the user will have the option to choose one of the methods to construct the phase angle master curve from which the damping ratio will be calculated.