© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

4.3.3.4 Pivoted Gyroscopic Accelerometers

Consider a cylindrical gyroscope spinning at an angular velocity w (radians/sec), with radius r and thickness h, whose center of gravity lies a distance L_p from a frictionless pivot point along the rotational axis. The pivot point defines the center of a spherical sensor grid of radius R (r² + 4h²)^1/2 having distinguishable grid elements located a distance Dx ~ 1 nm apart. Under uniform acceleration a, the gyroscope precesses around its spin axis at an angular velocity w_p = 2 a L_p / w r² = Dx / R t_meas, where t_meas is measurement duration. Hence the minimum detectable acceleration is

{Eqn. 4.16}

by comparing precessional motion against the grid.

In choosing an w with which to measure a_min, there is an upper and a lower limit. The upper bound is the bursting strength condition

{Eqn. 4.17}

where s_w is a safe working stress = 10¹⁰ N/m² (~0.2 tensile strength) for diamond¹⁰ and r is the density of the gyroscope material (~3510 kg/m³ for diamond). The lower bound is the minimum spin angular velocity below which the pivoted gyroscope cannot spin stably about the vertical axis and begins to wobble,⁴⁴⁸ thus ruining the measurement. The lower bound is given by

{Eqn. 4.18}

If w_max / w_min > 1, then nonwobbling spin velocities are available below the bursting speed of the gyroscope cylinder. Since this ratio scales with R, there is a minimum sensor size R_min below which no useful spin velocities are available. Combining Eqns. 4.16 and 4.18 and solving the quadratic in w gives

{Eqn. 4.19}

For t_meas = 10^-3 sec, R_min >~ 69 microns (optimum r = 48 microns, h = 25 microns); for t_meas = 1 sec, R_min >~ 6.9 microns (optimum r = 4.8 microns, h = 2.5 microns). Thus it appears that pivoted gyroscopic accelerometers likely will be employed only in devices >> 10 microns in size (but see Section 4.3.4.1). Microscale gyroscopes implemented in silicon have been discussed by Greiff ¹³⁸³ and others.⁴⁴⁷

Last updated on 17 February 2003