**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

**9.5.3.5 Hovering Flight**

In large helicopters, thrust is produced by imparting a downward
velocity to the mass of air flowing through the rotor, with lift proportional
to the change in momentum. In 1997, engineers at the Institute for Microtechnology
in Mainz, Germany, constructed a 3-cm long, 1-cm tall mini-helicopter weighing
0.3 gm with two blades turning at ~1700 Hz, and flew it to a hovering altitude
of 13 cm, then landed it safely. I. Kroo at Stanford University has built and
flown a "mesicopter" whose motor is 3 mm in diameter and 5 mm long, weighing
~0.3 gm, that turns the oddly-shaped rotors at ~50,000 rpm.^{3246}

As helicopter size shrinks through the transitional regime,
drag grows at the expense of lift and the rotorblade microhelicopter becomes
increasing inefficient. However, the viscous-lift helicopter (Fig.
9.22) is estimated to be able to hover by rotating its four wheels at a
frequency of n_{wheel} ~ 3 r_{nano}
g R_{wheel} / 32 h_{air} ~ 1 KHz,
taking r_{nano} = 2000 kg/m^{3},
g = 9.81 m/sec^{2}, and R_{wheel} = 10 micron.^{1982}

In the general case, the terminal velocity v_{terminal}
of a compact non-aerodynamic body of any size falling through fluid may be approximated
by setting:

and solving the resulting quadratic for v_{terminal}
(= v_{nano}), using F_{viscous} and F_{inertial} for
spherical bodies from Eqns. 9.89 and 9.90.
This velocity may then be used in Eqns. 9.88
through 9.92 to conservatively estimate F_{nano},
P_{nano}, and D_{nano} for hovering. Representative values are
given in Table
9.5, assuming flight in dry sea-level air and powerplant efficiency e% =
0.10 (10%). Hovering power P_{hover} scales as ~R_{nano}^{5}
for R_{nano} <~ 10 microns and as ~R_{nano}^{3.5}
in the R_{nano} ~ 0.1-1 mm range; for R_{nano} >~ 1 mm, aerodynamic
lift forces are available, the computation of which is beyond the scope of this
book. For R_{nano} = 1 micron in air, v_{terminal} = 120 microns/sec
(Section 9.5.3.2) and P_{hover} ~ 0.00005
pW (e% = 0.10 (10%)); for R_{nano} = 10 microns, v_{terminal}
= 1.2 cm/sec and P_{hover} ~ 5 pW (e% = 0.10 (10%)).

In the special case of flapping winged aerobots in the transitional
flight regime (e.g., 100 microns <~ R_{nano} <~ 10 cm) during
steady-state hovering, T. Weis-Fogh^{1578,1583}
calculates that the average aerodynamic power needed for small winged objects
to remain airborne is:

where r_{air} = 1.205 kg/m^{3}
at 1 atm and 20°C, t is a shape factor that equals
0.5 for a rectangular wing, 0.1 for a triangular wing attached at the base,
and 0.4 for a triangular wing attached at the apex; C_{D} = 0.07-0.36
for bats, birds, and insects with wing length of L_{wing} = 0.25-13
cm and wing width (chord) of w_{wing} = 0.7-55 mm (flight mass M = 0.001-20
gm); wingstroke frequency n_{wing} ranges
from 15 Hz at L_{wing} = 13 cm (large hummingbird) to 600 Hz at L_{wing}
= 0.25 cm; and stroke angle j_{wing} is the
angle subtended by each flapping wing in the flapping plane during a complete
stroke cycle, typically 2-3 radians (120°-180°).

Thus for example, the common honeybee *Apis mellifera*
(N_{R} ~ 1900) has M = 100 milligrams, L_{wing} = 1.0 cm, w_{wing}
= 0.43 cm, t = 0.27 (half-ellipse), C_{D}
= 0.09, n_{wing} = 240 Hz, and j_{wing}
= 2.09 rad, giving P_{hover} = 1 milliwatt with a dynamic efficiency
of e% = 30%*; during level flight at peak speeds near ~6 m/sec,^{739}
experimentally-measured honeybee metabolic demand is 20-60 milliwatts.^{1580}
Honeybees can also carry a ~40 milligram payload of honey. At the lower extreme
of the transitional flight regime, the parasitic chalcid wasp *Encarsia formosa*
(N_{R} ~ 15) has M = 25 microgram, L_{wing} = 620 microns, w_{wing}
= 230 microns, t = 0.50 (rectangle), C_{D}
= 3.20, n_{wing} = 400 Hz, and j_{wing}
= 2.36 rad, giving P_{hover} = 0.4 microwatt. Wingspeed v_{wing}
> 1.5 m/sec during the downstroke.^{1578}
From the conservative assumption that biological wing muscles are limited to
~200 watts/kg (~300,000 watts/m^{3}), Weis-Fogh^{1577}
estimates than no flying animal with a mass larger than ~100 grams can hover
continuously by means of wing flapping and wing twisting alone.

* A few other dynamic efficiency figures for
hovering flight include the hornet wasp (31%), hummingbird (51%), mosquito (70%),
fruit fly (95%), and butterfly (97%).^{1578}

For artificial nanorobotic flyers near the edge of the design
envelope, maximum wingbeat frequency n_{max}
<~ v_{wing} / L_{wing}. To avoid supersonic turbulence, v_{wing}
< v_{sound} ~ 343 m/sec in air at 20°C and 1 atm, giving n_{max}
<~ 10-100 MHz for L_{wing} = 3-30 microns and <~100 KHz for L_{wing}
= 3 mm. For each small-wing cycle at 100 MHz, the boundary layer after an impulsive
start is at most (modified from Prandtl^{1584})
d_{layer} ~ (h_{air}
/ r_{air} n_{max})^{1/2}
~ 0.4 micron ~ l_{gas} ~ 0.2 microns, the
mean free path of air molecules at 1 atm (Eqn. 9.23);
faster cycling would allow insufficient time for mechanical coupling to the
medium. Note also that v_{sound} is significantly slower than the torsional
deformation propagation velocity v_{torsion} ~ (G / r_{wing})^{1/2}
~ 12,000 m/sec along the wing structure, taking G ~ 5 x 10^{11} N/m^{2
}and r_{wing} ~ 3510 kg/m^{3}
for diamondoid materials. For comparison, the slowest insectile wingbeat frequency
is ~5 Hz for the swallowtail butterfly (*Papilo machaon*); the fastest
is ~1046 Hz for the tiny midge *Forcipomyia* in natural conditions, up
to ~2200 Hz at 310 K in laboratory experiments with truncated wings.^{739,2033}

Last updated on 22 February 2003