__Binary Flutter____ Solution for
Fluid Power. __** Please view in Internet Explorer to see figures**

**S.P. Farthing, Applied
Mathematician,Wing’d Pump Associates ****www.econologica.org**** ****spfd@cantab.net**** ****975 Tuam Rd. North Saanich B.C. V8L 5P2 Canada**

**Journal of Aerospace Engineering 2018**** ****Vol. 31, Issue 3 **

**Abstract-
**

*The stability of a foil with its
¼ chord center of pressure trailing a pitch axis sprung in heave is solved algebraically
to help design a fluttering windmill and perhaps watermill. Its flutter mode
and frequency/windspeed do not depend upon its total mass or spring rate. A ll
contours of this ‘reduced’ frequency in the pitch inertia & imbalance plane
pass through a nexus whose total inertia and imbalance are as if just the
virtual mass were at the ¾ chord point, with a mode of feathering in the
apparent wind at this aerodynamic center. The high frequency flutter amplitude
ratio is symmetric in pitch inertia about the nexus. Similarly from the second factor in its pitch damping, each contour passes through another
nearby simple point as if twice its Theodorsen factor times the virtual mass
were a ¼ chord divided by this factor behind the ¼ chord. So twice the virtual
mass at midchord gives the zero frequency inertia and imbalance “midpost” furthest away from
the nexus. Small trail makes the
imbalance greater at the midpost than the nexus so as to slope the zero
frequency line downward. Then the imbalance required for quasi-steady flutter
decreases with pitch inertia, even below nil beyond the nexus. Trail also bends
the gate of simple points to pass some low frequency contours very slightly
below the midpost to locally lower the flutter boundary. For an oscillating windmill the net virtual
mass reaction stiffens heave, opposed by the circulatory lift in flutter since
its pitch and heave are necessarily partly in phase. Such new results, and a water flutter demonstration show a practical
semi-rotary water blade would need a geared-up pitch flywheel for sufficient
inertia to flutter well. Whereas a wing
is so much heavier-than-air it has enough structural pitch inertia to flutter
and so pump easily. *

**Keywords: **mass ratio,
stability contours, Theodorsen function

**1. Introduction**

It was recognised early in
the study of aircraft flutter that its spontaneous phased oscillations of the
two ‘binary’ degrees of freedom were being powered by the airstream. Duncan (1948) even built a heaving ‘engine’
that articulated a balanced foil to pitch and heave
(or “plunge”) 90º out of phase to pedantically show this (and nothing
more). In fact to safely tap the highly
variable power of ambient flows requires exploiting both free amplitudes of
flutter (Farthing 2012). Our
FlutterWing’dPump (FWP) originated in 1978 after the 1976 BBC broadcast of
Pocklington School Young Scientists’
fluttering lab models promised
better wind waterpumping than
rotary multiblade windpumps, especially
for developing countries.

A fluttering windmill must be designed to
have a powerful instability to large amplitude over a wide range of moderate
windspeeds whereas
aircraft flutter simply mustn’t start before the never-exceed
speed. Previously unknown was 3D flutter
restabilisation in high winds, a key advantage of a fluttering windmill
hypothesised and verified on models in 1978,
then numerically in 1980, by a
full-scale FWP
in storms in 1990 and finally algebraically. (Farthing 2013).

As early aircraft
increased in speed,
flutter critical onset speeds were surpassed with many crashes
from the destructive oscillation of control surfaces unless aerodynamically and
mass balanced, or of wings unless torsionally stiff. Yet flutter has scarcely been a problem in marine
hydrofoils. The ratio of foil mass to
the virtual mass ** m** of the circumscribing fluid cylinder is much lower because of
the 700 times higher density. Solid
steel (Fe) hydrofoils would be very understressed but
still below unit mass ratio. Flutter calculations
for hydrofoils by eminent aeroelasticians (Ashley et al 1959) reinforced their
experimental evidence of a lower flutter limit of roughly equal real and
virtual mass. Their blades twisted
elastically behind the ¼ chord center of pressure so divergence frustrated
analytical solution and full understanding of the stability in water against
flutter.

So here the simplest non-divergent pure flutter linear model will be solved to clarify the mass ratio effect. To further motivate the algebra to come, the next section develops the conceptual niche of flutter pumping vs. heave engines vs. wind and water turbines.

Figure 1 Flutter Wing Pump schematic

**2. Flutter & Oscillating Capture of Wind and
Waterflow Kinetic Energy **

The ease of bird
flight, and the speed and leaping of fish, show a remarkable efficiency to
oscillating propulsion,
confirmed metabolically.
But for windmills, such an efficiency of power / drag/ flow speed ** V** bears only on total kinetic capture by an
array, and on capture per structural
cost of the drag (Farthing 2007), most
significant in water channels where the stream is not semi-infinite, and very
high forces produce the power at very low speeds. For the higher

As in Fig 1, a semi-rotary
3D oscillation of a wing from vertical about a low streamwise axis close
to the ground avoids, like the Vertical xxis wind turbine (Vawt), the high tower
mounting of the rotors, gearboxes and generators of the standard Horizontal
Axis (Hawt). Inverted in waterflow, these alternatives
would keep the bearings and power conversion out of the silty, if not salty,
bio-corroding water difficult for man to access. Ideally the Vawt’s power peak, narrower than the
Hawt’s, might capture a more acceptable
fraction of the tidal power than of the wider wind spectrum (Farthing
2009). But the fouling of a hydro-Vawt’s
blades and struts raises the foil drag ** D **to which its power is highly sensitive , and the straight blades of a (grid) synchronous
Vawt would vibrate badly from cyclic stall in peak tides. Whereas a semirotary
blade projects from just a bit of axle below a floating base, does not demand low

__Any__ reciprocating machine of
frequency **w** is stress
limited in its design flowspeed*
**V*_{d}*.***
**A wing length

In contrast in rotation, the main
inertial stress is centrifugal, or just benignly tensile in Hawt blades. So high rpm lowers the structural cost per
unit power and also the gear up to a generator to make __high __flowspeed
sites the most economic for wind turbines.
In high wind Vawts centrifugal dominance enables a catenary blade with
gradual stall and fewer struts to drag.

A slow oscillating prime mover for__
light__ flows could avoid a high ratio gearbox by reciprocating a pump,
whereas Hawt’s are very poor and Vawt’s worse at cranking piston pumps, especially
single-acting. Their ideal torque, varying as windspeed squared, severely mismatches the mainly angular torque
variation of cranking a single-acting pump with constant head. The difficulties are most intractable in the
more widely and rapidly variable wind with the pump constrained deep down a
well. (Dixon 1979) showed a rotary fanmill’s
useful pumped work is only 1/10 of the ideal capture of the annual wind energy
flux through such a swept area. Its 20
or more blades maximise the starting torque, but still not enough to turn over its
pump crank in the most efficient wind for its stroke. Wind tunnel flow visualisation and a flow
solution showed for the first time (Farthing 2011) how all of the kinetic
energy of the reaction flow to the torque is lost in the wake by centrifuging. Whereas an oscillating actuator approximates
a contra-rotating windmill (Farthing 2010) reducing this loss. The net ten fold was the primary motivation
to develop a pump oscillated by the wind. But many oscillation mechanisms would
be worse than Hawts in self-starting against the fixed pump head or handling
the mismatch of flowpower as the __cube__ of
flowspeed *V ** *vs.* *piston pump power as just the frequency **w**. With fixed
head, and articulated pump stroke, their frequency **w**** V ^{3}** to absorb the best fraction of the cubic
windpower but then inertial reactions on the articulation mechanism would
increase as

Instead at high *V*_{d}, flutter’s inherent dynamic balance of inertia, imbalances and flow
at near-constant frequency but variable amplitude can be non-linearly converted
into a highly variable pump stroke as in Fig 1. (Farthing 2014) composed full
amplitude equations of motion from the 2D fluid (Kochin 1964) and 3D rigid body
dynamics. Flutter’s non-linearities prove favourable to significant economic
extraction of moderate wind power for pumping by the “FlutterWing’d Pump”.

The wind doesn’t have any merciful ratio of
maximum to design average speed like a tidal flow. But it was speculated that as a resonance of
a fixed roll frequency with pitch frequency as windspeed , the flutter of 3D semi-rotary roll (Fig 1) might cease above
an upper critical ‘cutout’ windspeed. 3D
unswept wings of low imbalance indeed feather quickly and stably to the __true__ wind in a storm. As the cutout windspeed is approached, the
pitch amplitude decreases, containing the power and especially the downwind
thrust; whereas
to vary their pitch amplitude immensely
complicates articulations.

In very high winds 2D feathering to the
heave __apparent__ wind gives zero effective pitch stiffness and slow
flutter/ divergence. But the basic linear 2D pitch and heave flutter analysis
below does explain the ready start of the FlutterWing Pump in light winds
yet the stability of heavy ferrocement or even solid
steel blades in waterflow.

**3. FLUTTER MODEL & SYMBOLIC DETERMINANTAL SOLUTION**

Figure 2 : Section of Symmetric Airfoil Heaving across Wind and Pitching ahead of quarter chord

** c**, whose ¼ chord center of pressure (c.p.)
point trails by a distance

Pitch
elastic stiffness is not needed on the FWP and would interfere with
bidirectionality of a tidal flutter mill. It is also absent in cantilevered
‘spade’ rudders on boats,
if not on all-moving aircraft rudders . But the FWP pivot point
is sprung to heave cross-stream with coordinate* h*. Use the coordinate vector

(–** k^{2}A**+

with the real matrices, ** A** inertia,

[**A,B**]=A_{11}B_{22}+A_{22}B_{11}-A_{12}B_{21}-A_{21}B_{12} then **|****A|=½[A,A] **(2)** **

Then the nil
determinant in (1) for
neutral oscillatory stability in powers of ** k** expands to

** k^{4}|A|
-ik^{3} [A,B]
- k^{2}(|B|+[A,C]+[A,E] k_{n}^{2}) + ik (
**=0

where all the
crossed out terms will be shown to vanish in this simple case. For example **|E| **=0 is because the only elastic spring is **E**_{22} in heave. Equating imaginary odd power parts

*k*^{2}[**A,B]=
k_{n}^{2}[B,E]
**(4)

** **(or instead a
flutter/divergence ** k**=

** ** ** {|A| [B,E] -** **[a,E][A,B]} k^{4 } + {
[C,E][A,B] - (|B|+[A,C]) [B,E]}k^{2} **= 0 (5)

Either** k**=

* k*^{2}
= [A,B][C,E]- ( |B|+[A,C]) [B,E] * / * [A,B][a,E]-
[B,E]|A|

This ratio of the
difference of 4 way products has hitherto defeated meaningful analytic
treatment of binary flutter. Here each
product has an **E**_{22} factor so ** k** the conventional positive root is
independent of this heave spring

**
k^{2} = [A,B]**C

The naive Routh
criteria for stability of determinants not depending upon ** k** is lesser

**4. EVALUATION OF THE CROSS-DETERMINANTS**

(6) can be simplified by showing the flutter contours and mode depend on just inertia and imbalance, and evaluating each cross-determinant in its most convenient moment axis in terms of the key distance factors in the pitch damping.

For a thin airfoil, potential theory
gives a virtual ‘added’ mass of *m*= ¼**pr c^{2}** as in the circumscribing
cylinder of fluid (of density

Use ** m** to non-dimensionalise the

** **Let the spring
restoring force be ** S** per unit heave.

The heave real inertia and spring
difference is balanced by the virtual mass reaction and the circulatory lift *L** *acting at the ¼ chord point

* Sh +(p*-1)*mh”** *** = m^{
}**

* *

**-****g****‘ V **comes from a flow Coriolis
acceleration in the frame of the foil and negates the virtual roll inertia

** S**=

The pitch moment balance per unit span
about the pitch axis is ** m^{ }{**

** ^{ }**where

** ^{ }**Fig 2 shows the nominal apparent wind

** j=****g- h_{¾}’/V if**

*cq***g****‘**/** V** is sometimes called the effective camber

** U**=

When
**Q** are small and sinusoidal , the unsteady wake corrects these nominal
values by the Fig 3 wake factor function of reduced frequency

** **in terms of modified Bessel functions
of the second kind with

Figure 3 Theodorsen Function Real and Negative Imaginary Parts

So extend *h/c** *to the complex domain as (*Im*plied* Im*aginary *Im*
part of** e ^{i}**

**g**=(*Im*plied* Im*aginary *Im*
part of** e ^{i}**

Then the roll
equation (7) divided by *mV*^{2}e^{i}** ^{wt /}c** extends to

In the very high
wind limit of ** k_{n }**and

Henceforth for ** T=F**, the net pitch damping is

** G{-k^{2}j +4ikqyF+4eF}+H{k^{2}x-4ikeF}=0
**(12)

Aerodynamic balance ** e**=0 nulls the leading pitch stiffness so the dominant pitch balance is the
aerodynamic damping equalling the cross-forcing by roll via dynamic imbalance
or

[**A,E**]=**A**_{11}=** j
** [

(4) gives the flutter frequency** w ** only involving

(4) &(14) are also *k*^{2}{*pqy*-(*q+y*)*x+j*}
=*qyk*^{2}** _{n} **Substituting (8)

** **

Adding -** ec** times the lift Eqn(7)
to the moment Eqn (9) eliminates

*
***g”{ jc^{2}-ec**

This exchanges (complex) * L*

* * The eliminated ** eL** is often misidentified
as

*m***g****‘ Vc(e +½) **arises in the subtraction of a circulatory broadsiding moment in the
¾ chord apparent wind (at angle

*G***{ -k^{2}(j-ex)+½ik}+H{k^{2 }(x-ep)+ek_{n}^{2}}**

So**B,C**]=**|C|**=0 [**A,B]= ½ p+(1+4Fq)(ep-x)+4F(j-e x)**

As a recap below are the non-dimensional matrices with one of the first two rows used at a time. The second ¼ chord center of circulatory lift row was just used

** A**=

Whereas (13)** |A|= pj–x^{2}, **[A,B]=

were the general ** p** results with the first row moments
about the physical pitch axis . For the
numerator of (6)

** k^{2} (j-qx)(j-yx)
= 4eF(j-ex)+½x-2Fqy ** (19)

This may be the first time the high extent of algebraic cancellation
in the difference of products of determinants in (6) has been demonstrated. At
large inertia ** j**
,

or **0****»4 e (j-ex)+½/F-2qy
**so

From (4)** ,**
(13) and (19)

giving the flutter *V*^{2}*=**Sc*^{2}**/ mk_{n}^{2} **where

^{ }

**5. FLUTTER Contours
AND MODES**

Consider the
inertia and imbalance for just the virtual mass concentrated and relocated to
the ¾ chord point. Then**
sddx_{n}=j_{n}/x_{n}=q=e+½ vanishing** the numerator as
well as the denominator of (19) to solve it for all

(19) will also be
satisfied when *j=yx* instead of *j*=*qx*
intersects the RHS=0 line at * x*

To summarise, the nexus and gate and their modes are as if

point ** m** at Nexus

point 2** mF **at gate

*k*^{2}** **(21)

Recasting (19) in terms of these roots
as *k*^{2}*
***( j-qx)(j-yx) (**¼

Since the virtual mass
is not a point but distributed with the intrinsic virtual inertia of *mc*^{4}/32,
then to realise the midpost at ** k**=

From (** p_{0}**+1)

*q*^{2}-2*qx+*** j=0 **which radiates from
the nexus with half the slope of the
nexus ray. To the left at

In Eq. (19)* k*=0 gives the quasi-steady *F*=1
flutter boundary line
through the nexus and midpost*
*as

2*e*(*j*-*ex*)+¼*x*=*qy=*2*ej*+2*x*(⅛-*e*^{2}) with small *e*
slope about -8*e*. Comparing these
posts at *F*=1 *x*_{m}=2*e+*½=*e*+*x*_{n}* *and* j*_{m}=½*x*_{m}^{2} =(2*e+*½)(*e+*¼)
*<j*_{n}=*x*_{n}^{2}=* (e+*½)^{2} until ** e^{2}**=⅛ or

For a given ** e** and

Figure 4 plots the neutral ** k**
Theodorsen contours in the

For ** F**=½,

There is no sizeable quasisteady
zone to the left of the midpost above the lowest** j**=3/32 from the virtual
inertia about

Figure 5 Heave to Pitch
Amplitude Ratio ** r **vs Imbalance and Inertia Factors at

From
the pitch equation (12) ** H**/

{2

* *Tan**q
**of the phase lead **q**, or the slope in the complex plane of **G**/** H**
is 1/ 2

* n=**c***g _{0}**

This power per
pitch amplitude per naïve swept area is optimum at .27, little
changed using ** T**@

** H**/

Now fully introduce ** e**>0 with 4

** ** In Figure 7
for

The microzone of double ** k **is now extended below
and beyond the midpost. Such contours below the midpost would indicate flutter first appears at the lowest contour
and then increasing

From
(12) for ** e**>0 at

Figure 8 Heave to Pitch Amplitude
Ratio ** r** vs Imbalance and
Inertia Factors at

The righthand
contours basically reflect too, but the close and even crossing ** k**
and

Mass balancing without aerodynamic balancing
can even be insufficient to prevent flutter at high inertias. At the intercept ** x**=0 2

At** e**=√⅛=.35,

This detailled analysis
is a foundation for a further submission reaching exactness by including -*i G*(

**6. Flutter
Demonstration and Design
in Water**

In water the ** x** and

A trailing arm to achieve the desirable** j**
to the right of the nexus would be
massive. Instead extending a
long pitch axle above the roll axis out of the water to (lowbacklash) gear the pitch up in air by
ratio

Unless the pitch axis heaves perfectly
vertically, the weight of a dense blade
must also be considered. For instance
even a horizontal ferrocement hydrofoil would be severely weight imbalanced in
pitch and vertical heave. A semirotary
upright wing in air (Fig 1) has to be over-counterweighted in roll but its
tailheaviness forces pitch by roll [Farthing 2013] as well as by roll
acceleration. Given the low design water current speeds, *V*_{d}^{2}
<<** gR** this 3D gravity forcing effect is dominant in water even at
model scale. Conversely the weight of a
blade suspended in water is its own roll spring but must be nose over-counterweighted
(again better in air) for net noseheavy gravity forcing

A very long such pitch air arm allowed
a model test of the amount of extra inertia D** j**
required [Farthing 2015]. Fortunately
due to the phase shift

A floating tidal device would need to counter the
downstream moment of its underwater flow extractor. Fins angled upwards at the
anchored ends would ~~so~~ lift whichever end is upstream and depress
the opposite ‘stern’. To cancel the pump reactions, two blades could counteroscillate (assuming
sealife can avoid the shearing action.)
The difficult contralinking pitch
shouldn’t be necessary. But a pitch
starting track as in Fig 1 would
definitely be needed for both ebb
and flow~~each sign ~~of the tide not
being as shifty or gusty as a light wind.
Other design goals besides the essential stepup of each pitch to a
flywheel, are all bearings above or at
worst at the water surface and easy
winching the blades against each other to the surface for defouling. To realise
any such Fluttering (Hydro)Foil would
require very heavy safety-concious engineering and construction at a shipyard,
whereas the density change of 700 makes
the Fluttering Windpump construction lighter than light aircraft (Farthing
2014)

**7. Conclusions**

All flutter
contours radiate from the nexus of total imbalance and inertia about the
physical free-to-pitch axis as if from the virtual mass at the ¾ chord
aerodynamic center. They also pass through a gate curve from the nexus to a
midpost corresponding to twice the virtual mass concentrated at midchord at** k**=0.

Both are a substantial increase of imbalance moment ** x**
and especially inertia

The support of Gifford and Partners of Southampton, The Hamilton (Ontario) Foundation and the Science Council of BC are gratefully acknowledged.

__Nomenclature__

**r **Fluid density

**s** Material Fatigue limit stress

**g ** pitch angle

**g _{0} ** pitch angle amplitude

**j _{ }** Nominal Angle of attack of ¾ chord point

**q **phase lead**
**of pitch ahead of heave,

**G ** complex amplitude of
pitch

**w** circular frequency in radians of phase/unit time

**w _{n}** natural no-fluid
frequency in heave when

__A__* *non-dimensional inertia matrix

__B__* *non-dimensional aerodynamic damping matrix* *.

__C__* *non-dimensional
aerodynamic stiffness matrix

** E
**non-dimensional
elastic stiffness matrix

*c** * chord of foil

** D **Drag

D** **structural Material density

*ec*** **trail of quarter chord behind pitch axis in chords

** p** total mass/virtual

** g** acceleration due to gravity

** G **the neglected
imaginary part of the Theodorsen function

** h** heave of pitch axis

** h_{3/4}**
heave of the ¾ chord point

*H* non-dimensional
*h**/c* complex extended

*i** * square root of -1** **

*j** * pitch inertia/*mc*^{2}

** k ** reduced
frequency based on chord

*k*_{m }reduced frequency ** k >0 **of contour passing through midpost

** k_{n}
**reduced natural frequency

__L__* *circulatory lift, *L* its magnitude* *

*m** *virtual mass of foil/unit length*
*

*n** *Pitch amplitude*/ H
or *1

** p** ratio of total mass to virtual

** N** nominal apparent wind at ¾ chord point magnitude N

* Q*=(

*q** *distance
of foil ¾ chord from axis in chords

*r*** H** to pitch amplitude

** S **heave stiffness =force /

*F** * complex Theodorsen function of *k*

*U** * upwash or normal component of the apparent wind__ N__

* V flowspeed*

*V*** _{d}** design flowspeed

** w** live flow wing loading

** x** pitch
imbalance/

** y** distance of foil
midnexus from axis in chords

**’** denotes time* t* derivative of eg

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