**Nanomedicine,
Volume I: Basic Capabilities**

**©
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 (r^{2} + 4h^{2})^{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
r^{2} = Dx / R t_{meas}, where t_{meas}
is measurement duration. Hence the minimum detectable acceleration is

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

where s_{w} is a safe working
stress = 10^{10} N/m^{2} (~0.2 tensile strength) for diamond^{10}
and r is the density of the gyroscope material (~3510
kg/m^{3} 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

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