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The Dynamics of the
Flapping of a Wing

This material is on the dynamics of the flapping wings of a flying creature. It is a preliminary to a scale analysis of flying creatures; i.e., the weight of a creature is proportional to the cube of its scale, the wing area is proportional to the square of the scale, the muscle strength is proportional to the square of the scale and so on. How do the scale factors determine the feasibility of a creature flying?

Let φ be the angle one wing makes with the horizontal. This angle starts at zero for a flap and becomes negative under the combined torque exerted by the creature's muscles and the other forces on the wing. Let I be the moment of inertia of the wing. While flapping the wing is also subject to a torque due to the wind resistance.

Wind resistance is taken to be proportional to the velocity squared; i.e., αv2. The velocity at a segment of the wing at a distance r from the pivotal axis of the wing is rdφ/dt. Let b(r) be the width of the wing at r. Then the force due to wind resistance on the portion of the wing between r and r+dr is b(r)α(rdφ/dt)2dr and the torque is

rb(r)α(rdφ/dt)2dr = bα(dφ/dt)2r3dr

The wind resistance torque on the whole wing between r=0 and r=a is then

α(dφ/dt)20ab(r)r3dr

If the wing is rectangular and hence the width is independent of r the wind resistance torque reduces to:

αb(dφ/dt)2(a4/4)

The Dynamic Equation for a Flapping Wing

The general equation for a rotating body is

I(d2φ/dt2) = T

where I is the moment of inertia and T is the net torque on the body.

For a rectangular wing of length a, width b and thickness c the moment of inertia is:

I = ∫0aρr2bcdr = ρbc(a3/3)

where ρ is the mass density of the wing material.

The torque exerted by the wing muscles is the force exerted by the muscles times the torque arm which is some fraction of the wing length, say ha. The force exerted by the muscles can be taken to be a constant -F during the down flap and 0 for the retuurn motion. Thus

TM = haF

It might be thought that there would be a torque on the wing exerted by gravity and dependent upon the angle φ, but such is not the case for a free-flying creature. There would be such a torque if the creature were resting on a support but not for an unsupported creature.

The dynamic equation for the wing during the down flap is then

ρbc(a3/3)(d2φ/dt2) = -haF + αb(dφ/dt)2(a4/4)

This can be somewhat simplified by dividing through by ρabc, the mass of the wing, to get

(a2/3)(d2φ/dt2) = -h(F/ρbc) + (α/ρc)(dφ/dt)2(a3/4)

A further division by a2/3 yields

(d2φ/dt2) = -3h(F/ρa2bc) + (3/4)(αa/ρc)(dφ/dt)2

The force exerted by the creature's muscles F should be proportional to the cross section area of the muscles and hence proportional to a2. Thus F/a2 can be reduced to a constant, say β. The ratio (a/c) would be independent of scale and can be taken to be a constant, say γ.

The end result is a second order non-linear differential equation of the form

d2φ/dt2 = c0 + c1(dφ/dt)2

in which the coefficients c0 and c1 are dependent upon scale.

This is a first order differential equation in the rate of angular rotation ω=dφ/dt:

dω/dt = c0 + c1ω2

The above differential equation which is a type of Riccati equation has a solution but not always in closed form. The solution for ω gives a solution for φ which can be used to determine the lift on the wing.

The lift on the wing is the wind resistance force times cos(φ);

cos(φ)α(dφ/dt)20ab(r)r2dr

which for a rectangular wing is

cos(φ)αb(dφ/dt)2(a3/3)

This quantity integrated over a flapping cycle would give the average lift for one wing of a creature. The muscular force would be applied only from some intitial angle φ0 to a final angle φ1. Because of the lift depending upon the cosine of φ the flap get less efficient as φ deviates from zero.

A creature's weight is proportional to the cube of its scale and thus would be proportional to a3. Dividing the above expression by a3 yields an expression which is proportional to b(dφ/dt)2. If (dφ/dt) were independent of scale the analysis would imply that larger flying creatures get more lift per unit weight than smaller creatures, a result that is not likely to be correct empirically.

Empirically it is clear that (dφ/dt) does depend upon the scale of a flying creature. A hummingbird's wings flutter at a rate that would be impossible for a condor or even a sparrow.

The solution for ω when ω(0)=0 is

ω(t) = (c1/c0)1/2tanh((c0c1)1/2t)

The analysis will be continued at a later time when more information on the dependence of the solution of the dynamic equation for φ on scale can be obtained.

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