Parametric Resonance in Microcantilevers with Application in Mass Sensing

1.1 Characteristic phase dependance between direct forcing and parametric amplified forcing phase. Plotted is gain vs. phase at resonance for a 2.1 Simulated response of a parametrically resonant system under two driving conditions. The drive signal is shifted by pi-radians. The physical response, displacement is plotted here, is identical as driven by the two signals. Thus, phase shifting the drive by pi does not effect the system response.. Simulated observations of the three regions of a parametric resonant system for a negative cubic stiffness coefficient (γ < 0). [a] Spectrum of stable and unstable solutions of parametrically resonant system using the parameters in table, and the regions of stability segregated by the two bifurcation points of the system. [b] Phase space of region 1, shows 3 stable points. One stable point is at origin (no motion) and the other are two are symmetric about the origin. [c] Phase space of region 2. Origin unstable, symmetric stable solutions. [d] Time series growth of solution amplitude to stable point with parameters from c. Initial conditions [0. Changing the polarity of the nonlinear parameter γ does not effect the linear stability region. The system on the left is a 'softening' system, that is it has a negative cubic nonlinearity and large amplitude oscillations cause a lessening of the linear stress-strain relationship. The system on the right is 'hardening'. Parameters used in simulation in-5 2.6 A comparison of 'duffing' oscillator (top) and parametrically excited oscillator (bottom) for increasing drive amplitude and of equal parameters. Notice that using large forcing can increase sensing amplitude by orders of magnitude.