Fig. 2: Experimental realization of higher-order singularities in micro-electromechanics.
From: Higher-order singularities in phase-tracked electromechanical oscillators
a Two near-degenerate in-plane standing-wave modes of a microelectromechanical disk resonator are used to realize the scheme in Fig. 1a. b Experimental setup. Mode 1 is driven differentially by a force \({F}_{0}\cos ({\omega }_{{{{{{{{\rm{d}}}}}}}}}t)\), while mode 2 remains unexcited. The device is mounted on a rotating rate table to introduce an out-of-plane rotation \(\overrightarrow{\Omega }\). The charge amplifiers transduce the antinodal displacements of both modes, q1,2, which are then recorded and demodulated by a lock-in amplifier, to yield their amplitude and phase responses. The PhT condition is enabled by activating the PLL to lock the phase of mode 1 in quadrature, θ1 = − π/2. The degeneracy condition Δω can be adjusted by applying an electrostatic tuning voltage Vt to the antinodal electrodes of mode 2. c Natural frequencies of the modes, ω1,2, versus tuning voltage Vt. d–f Experimental frequency responses of the amplitude and phase versus the angular velocity Ω for the degeneracy conditions Δω ≈ 0 (d), − γ (e), and γ (f). The dot-dashed curves in the ∣q1∣ responses represent the eigenfrequencies. Colored contours in the θ1 responses indicate the θ1 = − π/2 PhT frequency \({\omega }_{{{{{{{{\rm{d}}}}}}}}}^{*}\), confirming (d) the “pitchfork” bifurcation and (e, f) the saddle-node bifurcations. g PhT frequency \({\omega }_{{{{{{{{\rm{d}}}}}}}}}^{*}\) measured by the PLL versus the angular velocity Ω at degeneracy. The error bars are the standard deviation. The colored curve is the theoretical result. h PLL measured \({\omega }_{{{{{{{{\rm{d}}}}}}}}}^{*}\) when the tuning voltage Vt is adiabatically swept at constant angular velocities. The blue dashed (red solid) curves depict the Vt-increasing (decreasing) sweeps, illustrating singularities and hysteresis if Ω0 > Ω0. The gray surface is the theoretical result. i Singularities projected onto the Vt-Ω plane. The white-faced points (light blue curves) are experimental (theoretical) data.
