# Wingspan formula for an Ornithopter

How do i figure out what the wing area/sqft needs to be for a functioning wing-flapping ornithopter? Lets say for example there is 500# of force moving the wings up and down, and the wings and passenger combined weighs 250#. What is the minimum needed wingspan/area to climb through the air?

• "Force moving wings" has to lead to a net downward momentum greater than the mass*speed you're trying to lift. Total wing area is irrelevant other than for efficiency of force-to-air movement. Nov 29, 2016 at 16:08

The graph below provides areas for living and man made flying machines - and this can be seen to be a quite good log-log fit of mass and wing loading. See his text for comment on the departure of Humming birds from the continuum mapped by most other birds.

The graph suggests wing loadings of 30 to 80 kg/m2 as mass rises from 100 to 300 kg.
Or about 3.3 m2 at 100 kg and 3.75 m2 at 300 kg.
ie close to constant across your likely mass range.

I have added the red lines and white text.
Not that he works in Newton and I have roughly converted to kg.

Flapping rate - "flat plate lift":

Based on the drag equation, and miscellaneous assumptions (which I can explain if of enough interest) I estimate a required "flapping rate" of about.

Flaps per second ~~~~~>= 4 x sqrt(Mass kg / Area m2) / stroke2
= Fps = 4.sqrt(M/A)/stroke2. Stroke in meters.
e.g. for 4 m2 wing area, 160 kg load, 1.0 metre stroke:
Flap rate >= 4 x sqrt(160/4)/1.0 = 25 flaps/second

This is based on the force on a flat plate when waved in air at sea level in one direction over a linear distance = 'stroke' art a rate of 'flap rate' per second with drag in one direction and mone in other, and high drive:return time ratio.
(Using drag equation - Force ~~~= 0.6 x Rho x Cd x A x V2. )

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Ornithopter wing design algorithm:

MATLAB code for the wing design is given here on this page.

Due to 'potential navigation difficulties' due to language used I have appended the simple sample code at the end of this answer.

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The structure of bird wings with applications to ornithopters

The first animated GIF is of arguable usefulness but a great thought starter - Wheeee!

See also Aerodynamic ornithopter fixed wing considerations - all sorts of technical goodies.

Real world non-ornithopter but relevant man powered flying machine

The man powered flying machine in the videos below is not an ornithopter but maintains takeoff & flight lift based solely on "man powered" surfaces and the same basics can be applied. In the video example there are 8 unidirectionally driven lifting surfaces. An ornithopter could conceptually drive the same area at about the same mean forwards velocity but with powered downbeat and (probably) liftless upbeat to produce the same net effect. For now, see this most lovely 1m30s video Also (slightly longer) https://www.youtube.com/watch?v=syJq10EQkog
and slightly longer 'making of'

Some hard data and real models here.
In Iranian (Persia)(Farsi?)- web translate works OK. Aerodynamic design wing ornithopter

http://s6.picofile.com/file/8266611400/52.JPG

His table of wing functions vs wingspan is liable to be of interest.
This gives values for all birds for (as well as other hings) power, wingspan, wing area, power in terms of body mass

Span = 1.17 x m0.39
Power = 10.9 x m0.19
Beat rate = 3.98 x m-0.33

Plugging ornithopter scale data into these suggests I'm missing something. TBD.

http://s6.picofile.com/file/8266611518/55.JPG

MATLAB Ornithopter wing design code **from **

Pasted as is - tidyfy as desired or see source page.

%% ---------------Ornithopter Wing design -------------------- %
--------------------- wwww.Asec.ir -------------------------

clc;clear;close all %% Fase ONE..... m=0.4; AR=3; W=m*10; W_S=30.6*W^(1/3) S=W/W_S span=sqrt(AR*S) a=span/2; b=S/pi/a
f=3.98*m^(-0.27) x=0:a/20:a; y=b*sqrt(1-(x./a).^2); Xc=[0 a];Yc=[0
0]; X=[0 x];Y=[0 y]; fill(X,Y,'b','LineWidth',3);hold
on;plot(Xc,Yc,'r','LineWidth',3);axis equal text(a/2,3*a/4,'
\leftarrow MAC','FontSize',14)

• For clarification, I want to takeoff at a hover, not just by moving quickly forward. The wings, from an extended downward position, are contracted inwards to reduce air resistance, the wings are pushed upwards, then fully extended for the downstroke. The wings are also on a ball and socket joint, so taking off upwards, they can be flapped at a tilt to make better use of the foil. Nov 29, 2016 at 8:10
• @E.Pence Understood - And/but - your wings MUST do something like a bird does - you cannot wave in an arbitrary manner and expect optimum results. UP stroke MUST have minimal drag and DOWN stroke maximal. Did uou watch the video(s). In an copter the blade moves forwards against then air continually. In an ornithopter the blade/wing feathers and moves back to original position with minimal drag. I saw model paper and plastic ornithopters in China with wind up rubber band propulsion system. Great fun and demonstrate practicality - of sorts. Large area and slow flap liable to be optimal. Nov 29, 2016 at 8:23
• Nov 29, 2016 at 8:29
• Serious formulae and models here - in Iranian. Google translate seems OK. asec.ir/… Nov 29, 2016 at 8:31
• ^^precisely what i meant to say, i guess i probably didnt explain it very well. The wings are bowed in on the upstroke, and extended on the downstroke, unlike with all ornithopter designs i have seen which are nothing more than stupidly long wings going really slowly up and down. Nov 29, 2016 at 8:35