The Intersection of Noise, Amplitude, and Nonlinearity in a High-Q Micromechanical Torsion Pendulum

We derive a nonlinear equation of motion for a chip-scale pendulum comprising a thick plate suspended from a tensioned nanoribbon. Recently, we explored the use of such a device as a clock gravimeter, exploiting the parametric coupling of its frequency to the local acceleration of gravity and demonstrating micro-g resolution with a silicon nitride prototype. Here we consider the restoring torque arising from the mid-plane stretching of the nanoribbon, finding it is a hardening spring that can be used to counteract the softening of gravitational torques, reducing parametric frequency noise and extending the range of isochronous pendulation. Using the method of multiple scales, we predict that parametric frequency-amplitude coupling can be driven to zero by exploiting fabrication tolerances available using modern nanolithography.