content_image/370721.jpg content_image/370722.jpg content_image/370723.jpg content_image/370724.jpg content_image/370725.jpg content_image/370726.jpg
ish Ministry of Science and Technology through the projects NAN2004-09183-C10-3 and MAT2006-12970-C02-01. 

## VIII. APPENDICES 

## A. Appendix A 

In the harmonic approximation, the fundamental equation describing the dynamics of a SFM-system is that of the forced harmonic oscillator, m ¨ a ( t ) + γ ˙ a ( t ) + c a ( t ) = F ( t ), where c is the force constant of the system, m its (effective) mass, γ the constant describing the damping in the system and F ( t ) the external force driving the oscillator. With the definitions ω$_{0}$ = ( c/m ) 1 / $^{2}$, Q − 1 = γ/ ( mω$_{0}$ ) = ω$_{0}$γ/c , and assuming a harmonic driving force F ( t ) = m a$_{0}$ω 2 0 cos( ωt ) = m a$_{0}$ω 2 0 Re( e $^{iωt}$) , where a$_{0}$ is a displacement determined by the driving force ( a$_{0}$ = F (0) /c ), this equation is transformed into 

<img src='content_image/370721.jpg'>

Note that, in order to avoid recurrent 2 π factors the angular frequency ω = 2 πν will be used here instead of the frequency ν as in the main text. For the steady state motion this equation can be solved algebraically with the classical ansatz a ( t ) = Re( A ( ω ) e $^{iωt}$): 

<img src='content_image/370722.jpg'>

from which the complex amplitude A ( ω ) is determined as 

A dimensionless (but complex) mechanical gain 

<img src='content_image/370723.jpg'>

<img src='content_image/370724.jpg'>

can be defined so that A ( ω ) = a$_{0}$G ( ω ). For the discussion that will follow, it is more convenient to describe the complex amplitude A ( ω ) in Cartesian coordinates: 

<img src='content_image/370725.jpg'>

<img src='content_image/370726.jpg'>

