This
post is the continuation of Angle of attack windvanes post
. I will continue here to introduce the
math that underlies my DIY
vane making. I
will expose some preliminary
information that should be well aware during
the design synthesis, the post is focused
on the example AOA vane sideslip angle
can be measured following the same scheme.

Assuming
a second order system as
model for the
dynamic of the wind vane
yield to consider
two performance parameters
in early design
stage; the first is the resonance frequency
and the second is damping.
At first bearings or other forms of
friction effects on
the main shaft are completely neglected. Resonance
frequency knowledge
is used to verify
also that the
windvane is non excited by other mechanical devices, as depicted in
previous post. If a variable gust is used to
excite/move the vane, to have a meaningful
position reading is paramount that gust frequency components do not
exceed nor approach
to resonance frequency, in such a way the transfer function will have
a unity magnitude. Damping indicate how
much fast the output oscillation
magnitude decrease with time. Look to the
following picture that illustrate
a second order system time response with
different natural frequencies
and damping values. Our windvane will be
underdamped, hence damping value will be
enclosed in the (0,1) interval.

Following equation
define a second order transfer function and common notation, since
our vane will be underdamped the typical time response will be
oscillatory with a time decay exponential term that reduce the
oscillation magnitude in time.

$$ H(s)=\frac{\omega^2_n}{s^2+2\zeta\omega_n s+\omega^2_n}$$

\(\omega_n\)
is the natural frequency

\(\zeta\)
or zeta is the damping

**EQ 7.1**Standard second order transfer function .Transfer function dc gain is unitary.\(\omega_d=\omega_n\sqrt{(1-\zeta^2)}\) is the damped frequency,oscillation in time occur at this frequency

**EQ 7.1b**Damped frequency definition

**AOA7.1a**Second order system step response*Damping value as follow Black 0,15, Blue 0,2, Green 0,4, Light blue 0,8.*

**AOA7.1b**Second order system step response*Natural frequency wn value as follow Black 10, Blue 20,Green 100.*

Figure inspection shows that for a good system behavior, good for a fast and stable measurement, we need a high value for both resonance frequency and damping; you find a scilab file to play around with parameters here. To increase the windvane resonance frequency, as per many mechanical systems, is needed to decrease the inertia. Our vane is statically balanced with a fore mass, overall weight may be decreased augmenting the distance of the counterweight from the rotation axis and decreasing the distance of the fin from the same axis. Decreasing the weight in such a manner will increase the whole wind vane inertia and lead to a reduction of the resonance frequency. On the other hand using a longer fin arm will increase damping of the system that is a

*desiderable behavior as per reference here eq.6.*

So we have indeed two parameters that tow our design in
opposite directions. If an extremely fast dynamic response is needed
the ideal design is a leading edge windvane, it's a vane that pivot
directly on the leading edge of the fin with no counterweight on the
other side of the rotation axis. Searching for high values of damping
lead to a classical, long arm, wind vane.
Since there is no a dominant argument to
chose those values only coarse constrains can be given, regarding
damping should be advisable to choice a value greater than 0,15
to avoid brutal oscillations during operation. A
damped system will oscillate at the damped
frequency wd of
Eq.7.1b as per figure AOA7.3, when damping is low wd value is pretty
close to the natural frequency value.
Natural frequency,
to warrant unity gain, should be two times the maximum frequency that
it's intended to measure with the vane. As
per previous post resonance frequency should be lower than \( 0,3Airboom_resonance Hz \) ; this value is
at last a good first guess
frequency, the more distance between
resonance frequencies the better.

In
a typical operating scenario
the vane deal with a rather constant wind
speed tanted by gusts, for design and
simulation purpose it's possible to use Dryden wind model
or FAR
FAR2 as
well of any other
documented wind gust model; such
a models are
often available as a library block on many simulation packages.

Dryden
model is based on a stochastic representation of wind gust, passing
through the autocorrelation function yield to the definition of a
filter that once feed with a with noise signal can generate wind
parameters, such parameters are frequency dependent. Both the models
are not intended to provide a very accurate result for limited
altitudes. Anyway they still useful because define a possible shape
for a wind gusts and provide a base scheme to compare different vanes
performances.
I'll
use a FAR flavor model for preliminary design stages and during the
simulation phase I'll consider also a Dryden model.

Gust
model will be used to define the
wind magnitude shape on
the z axis, the wind is assumed to be
blowing exactly perpendicular to
the ground and
pointed to the
airplane; using a North East
Down reference
frame
the wind is coming from the Down axis.

Recalling
that wing section relative speed is equal to vector sum of ground
speed and wind speed we get that if the vehicle is in level flight a
variation of vertical wind speed lead to a variation of AOA, if the
vertical speed rise the AOA is increased and vice versa.

Refer
to the following figure as 1-cosine FAR
gust shape, the
wind components are assumed to act independently
and only z component is shown, customize
the model parameter with your test conditions, download
the plot file here.

**AOA7.2**1-cosine FAR flavor gust shape ude=15m/sRight gusts parameters does not exist, during design phase a worst scenario approach will be applied, considered scenario will be tied to pitot intended use and flying Platform.

The
length of gust give us and indication of what is the impact of vane
dimensions on the accuracy, to make a long story short it's not
possible to measure a 10 cm gust with a 25 cm vane; with this
dimensions
the tip of the vane can be considered immersed in one gust while
the
tail is
into another.
It
is
useful
to
define
the minimum steady
wind path length
required for
a stable AOA reading.
This parameter, have
been defined
by many authors, is
called in
the reference decay
distance and can be calculated for a given windvane, it
is
the distance that the instrument should travel into the gust to reach
a given percent of the final AOA value starting from a
different AOA value. You
can visualize a typical windvane(compare
with a
wind
tunnel release
test
from 1974, JAMIES
T. KARAM, JR.,
TECNICAL REPORT AFIT TR 74-8 figure 19 pag 26 ) time response of a release test in the following figure.

**AOA7.3**Typical Windvane simulated release test, tl or*Decay time is depicted as the time takes to the vane to go cover the 90% of the travel between initial and final value. tl the equivalent in time of decay distance*

*Damping 0,15, wn 10 1/s \(w_{d}\) 9,88 1/s*

*Oscillation period \(a=\frac {1} {wd/2\pi}=0,635 s\)*

An ideal airborne vane have a decay distance equal to zero, for the proposed vane a 50 m value it's a good initial value. However this parameter lead only to a static performance indication and do not correlate with dynamic behavior. Referring figure AOA7.2 The 1-cosine gust have a maximum derivate \(maxp = ude/2\)

To
bound the vane performance it's possible to calculate the time response of
our system to a ramp

$$ u(t)=maxp\cdot t$$

From
this reference table
3we get the time response of a second order
underdamped system to a ramp input; the time domain output is the sum
of a ramp an offset and an oscillating term.

**EQ 7.2**Standard second order system unitary ramp time response
$$ y_s(t)=\frac{1}{omega^2_n}\bigg[t+\frac{e^{-\zeta\omega_nt}}{\omega_n}\bigg(2\zeta cos\omega_dt+\frac{2\zeta^2-1}{\sqrt{1-\zeta^2}}sin\omega_dt\bigg)-\frac{2\zeta}{\omega_n}\bigg]$$

The
third term, a pure offset error, decrease if natural frequency is
increased; second term is dominated by a time constant of magnitude
\(\frac {1} {\xi w_{n}}\) hence the oscillatory behavior will be quenched also by a
\(w_{n}\) increase.

Before
proceeding I summarize the simplifications and the parameters that
will be further employed.

*Simplifications*

-Sensor is ideal, only the mechanical part or primary is under analysis.

-Aerodynamic
interaction between different part of the vane are neglected

-Bearing
friction is considered at first negligible, no stiction phenomenon.
The bearing impact will be accounted in a secondary phase.

-Model
yield for little AOA values, wind tunnel test confirm that linear
model doesn't yield for high AOA values

-Gusts
models not conceived to be accurate for very low altitudes, let say
under 100m, anyway gust shape is taken as reference shape for
performance comparison purposes

-Windvane
support is 'stiff enough' and so considered as fixed, as previously
discussed is primary that no mechanical interaction occurs between
this two components. Also any other form of mechanical
interaction/resonace with flying components have been checked and/or
excluded.

*Performance parameters, notation, design goal initial values for example vane or main constrains*

-Operating
speed range, (10m/s,60m/s) Derived from intended use / Flying
platform specs

-Operating
AOA range, (-30°,+30°) Derived from intended use / Flying platform
specs

-Vane
static measure offset, statico, \(|statico|<0,1°\), Can be calculated
on the application requirements basis

-Vane
Resonance frequency , \(w_{n}\) , 20 Hz, A higher value reduce the ramp
tracking error

-Vane
Damping coefficient, \(\zeta\), >0,15, \(\frac {1} {\xi w_{n}}\) should be minimized
for good ramp response, a good high damping warrant a fast response
to measure variations

-Decay
distance, \(utd\), \(utd<50m\)

*Here below a list of geometric parameters, consistent with*reference Fig.1

-Fin
aerodynamic center distance from rotation axis \(rv\)

-Counterweight
distance from rotation axis \(rw\)

-angle
of attack, AOA, in the reference greek letter beta is intended for
yaw since they are considering a meteorological windvane, for our
application as usually denoted I will use the greek letter alfa \( \alpha \).

*Relevant aerodynamic parameters are*

-Aspect
ratio of fin \(A =b^2/S\) , where \(b\) is the fin fullspan and \(S\) the fin
surface.

-Coefficient
of lift and drag \(Cl(AOA)\) and \(Cd(AOA)\).

Now
that the windvane design requirement problem have been introduced and
bounded it's necessary to explicitly correlate the vane geometric and
aerodynamic characteristics to performance parameters.

To next post

JLJ