**Equation of Motion of an Individual Kink in the Geometric Kink Chain**

**Coordinate Transformation: Global (x) to Local Coordinates (u) **

Assumption: Nearest-Neighbor Interaction between the Kinks

The time and space variables are normalized as;

with respect to the driving frequency and;

with respect to the distance travelled by a spread-out interstitial during one period of the motion.

**Calculation of Dynamical Internal Friction Coefficient**

where

**Wdisp:** Energy Dissipated per cycle by the total kink population

**WE:** Maximum elastic energy stored in the kink chain during one cycle of external excitation

**Dissipation energy;**

Due to spread-out interstitials

Due to decorator point defects

and,

where is the macroscopic relaxation strength of the sample. This factor is previously described by **Granato-Lucke, Schoeck, Seeger, and Ogurtani.**

Finally the following equations were used to calculate the internal friction. For the Parent Damping Peak:

and

For the Decoration Damping Peak