© 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/m³, g = 9.81 m/sec², 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:

{Eqn. 9.93}

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_nano⁵ 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:

{Eqn. 9.94}

where r_air = 1.205 kg/m³ 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/m³), Weis-Fogh¹⁵⁷⁷ 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 Prandtl¹⁵⁸⁴) 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 10¹¹ N/m² and r_wing ~ 3510 kg/m³ 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