Estimate of the Damping Force Exponential Coefficient for an Oscillating Beam

For many dynamic systems damping/dissipative forces (DDF) are important. These forces are generally modeled in the equations of motion by terms linear in the ``velocity." An example is the standard damped, linear harmonic oscillator. However, for complex systems a broader range of functional forms is required for the associated DDF. If the fact that such systems only oscillate in a finite number of cycles is taken into account, then the leading term of the DDF is proportional to $v^\alpha$, where $\alpha$ lies in the interval $(0,1)$. We present preliminary experimental results, for a vibrating beam, implying that $\alpha\sim 0.9$. To obtain this value we derive a theoretical relationship between the damping time and the ``initial amplitude" of the beam, a relationship which does not depend on knowing {\it a priori} the exact equations of motion. Our findings are relevant for the study and analysis of vibrations in carbon nano-tubes and graphene sheets.