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 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 $^{iωt}$): from which the complex amplitude A ( ω ) is determined as A dimensionless (but complex) mechanical gain 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: ¨ a ( t ) + ( ω$_{0}$/Q ) ˙ a ( t ) + ω 2 $_{0}$a ( t ) = a$_{0}$ω 2 0 cos( ωt ) Re $^{{}$− ω $^{2}$A ( ω ) e iωt + ( ω$_{0}$/Q ) iωA ( ω ) e iωt + ω 2 0 A ( ω ) e iωt $^{}}$= Re( a$_{0}$ω 2 $_{0}$e $^{iωt}$) A ( ω ) = a$_{0}$ 1 − ( ω/ω$_{0}$ ) 2 + i ( ω/ω$_{0}$ ) /Q (16) G ( ω ) = 1 1 − ( ω/ω$_{0}$ ) 2 + i ( ω/ω$_{0}$ ) /Q (17) X ( ω ) = a$_{0}$ 1 − ( ω/ω$_{0}$ ) 2 (1 − ( ω/ω$_{0}$ ) $^{2}$) 2 + (( ω/ω$_{0}$ ) /Q ) 2 (18) Y ( ω ) = a$_{0}$ ( ω/ω$_{0}$ ) /Q (1 − ( ω/ω$_{0}$ ) $^{2}$) 2 + (( ω/ω$_{0}$ ) /Q ) 2 (19) 14