Numerical Simulation of Von Karman Street generated by a strut
This post is the second part of a series; the first part can be found here. It is an introduction to the math related to this DIY wind vane. Background information will be presented, essential to the design process. This post is focused on the design of an Angle-of-Attack vane. A sideslip angle design follows the same principles.
Modeling the dynamics of the wind vane a second-order system as model, we end up with two significant parameters of the system: resonance frequency and damping. Initially, bearing and other sources of friction on the main shaft will be neglected. A known resonance frequency value is used to verify that the wind vane is not excited by other mechanical devices, as depicted in the previous part. Wind gusts are generally an intended source of AoA readings, but we must ensure that their frequency content does not approach nor exceed the resonance frequency, so as the frequency response of our system has unity magnitude. Damping indicates how faster an excited oscillation decreases with time. The following picture illustrates the time response of a second-order system, while varying the value of damping. Our wind vane will be underdamped, hence its damping value will lie withing the (0,1) range.
The following equation defines a second-order transfer function using common notation. Since our vane will be underdamped, the typical time response will be oscillatory, with an exponential time decay term which reduces the oscillation magnitude in time.
is the natural frequency
or zeta is the damping
Equation AOA 1 - Standard second-order transfer function. Transfer function DC gain is equal to one.
is the damped frequency, oscillation in time occur at this frequency
Equation AOA 2 - Damped frequency definition
Figure AoA 5a - Second order system step response
Damping values as follows Black 0,15, Blue 0,2, Green 0,4, Light blue 0,8.
Figure AoA 5b - Second order system step response
Natural frequency value as follow Black 10, Blue 20,Green 100.
Inspection of the previous figures reveals that for good system behaviour (fast and stable measurements) we need a high value for both resonance frequency and damping. A SciLab file available for experimenting with parameter values can be found here. To increase wind vane resonance frequency, much like many mechanical systems, inertia decrease is required. Our wind vane is statically balanced with a fore mass and the overall weight can be decreased by enlarging the distance of the counterweight and decreasing the distance of the fin from the rotation axis of the vane. On the other hand, using a long fin arm will increase the damping of the system which is an also desirable characteristic, as is also described here, eq.6.
Thus, we end up with two conflicting requirements in our design phase. If an extremely fast dynamic response is needed, the ideal design is a leading edge wind vane, which is a vane that pivots directly in the leading edge of the fin, with no counterweight. If high values of damping are required, the classical long-armed wind vane is more promising. Since there is no dominant argument when choosing those values, a value greater than 0,15 is a good start, in order to avoid uncontrolled oscillations during operation. A damped system oscillates at the damped frequency
Wd of eq.2, as per figure AoA 5b.
When damping is low, the damped frequency value is close to the natural frequency. Natural frequency should be two times the maximum frequency of the measured quantity, in order to ensure unity measurement gain. Also, as has been said in the previous post, vane resonance frequency should be lower than 0.7 times the Airboom resonance value, or at least, this is a good first estimate; the more distance between the two resonance frequencies the better.
In a typical operating scenario, the vane will have to deal with a constant wind, with gusts superimposed on it. For design and simulation purposes gust models are available, such as the Dryden, FAR and FAR2 models. Wind gust library blocks also exist in many simulation packages.
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