## Thursday, August 1, 2013

### DIY Angle of attack windvanes part 2

This post is the continuation of Angle of attack windvanes . 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  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/s

Right 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