Keel profiles, blunt trailing edge and xfoil/XFLR5

The same would happen in 3D... that's how you get a humming keel/centerboard/rudder: When the frequency of the periodical vortex shedding gets close to the natural frequency of the centerboard, the appendice resonates and starts to vibrate, causing the humming. Judging from the sound that would be maybe 50-70 Hz? The vortex shedding frequency depends on flow speed (Re number?), so you need more speed than 6 knots to make a 1 m chord keel hum. Whether the vibration increases or decreases the drag, I don't know, I guess the general opinion is it increases the drag? At least it disturbs the helmsman so much that you want to get rid of it.

A similar phenomenon causes the shrouds to "sing" in the marina. Since the natural frequency of a taught wire depends on the tension, one would imagine that less tensioned shrouds will start to sing in lesser wind, but they will sing at a lower tone that highly strung shrouds (?)

How would you calculate the natural frequency of a rudder? Is it a cantilever beam that starts to vibrate all the way from the root (probably not), or is it just the trailing edge (or part of it) vibrating?

 

I would say that cantilevered beam with fixed root and rotational DOFs included is a correct way to model the problem. There will be an infinite number of natural frequencies, mostly coupled. Which ones will be involved in the "singing" phenomena depends on the frequency of the vortex shedding, which (for a given trailing edge width) in turns depends on the waterflow speed and angular position of the rudder. The angular position will at least partly depend on the vibrational torsions around Z-axis and local translations along Y axis, and so on. An iterative task, not at all trivial.

 

Fixed root and rotational DOFs, yes, that would seem to make sense.

 

Rudder shaft fixed at the root, translations allowed along Y axis and rotations around X and Z axes. The other DOFs are irrelevant for the problem.

 

My experience with "singing" centerboards has been that the pennant (rope/wire used for raising the centerboard) is usually the cause of the singing.

For a rudder I would not neglect rigid body motion forced by the alternating pressure from vortex shedding. The structural natural frequency would be zero, but there may be a non-zero hydrodynamic modal frequency when the hydrodynamic forces and moments (without vortex shedding) due to the deflections of the rudder are added. The combination of frictional and hydrodynamic damping may be sufficient that the rudder doesn't oscillate on it's own but could with the addition of the vortex shedding.

Also keep in mind that it isn't always the first mode / fundamental frequency which causes problems. I've seen vehicle radio antennas which would repeatedly and visibly vibrate in second mode at certain speeds due to vortex shedding, but never vibrated in the lower frequency first mode.

 

Vortex shedding frequency correlates with length and flow speed by the Strouhal number St:
St = frequency * characteristic length / speed
Whether vortex shedding occurs and if it does the type of shedding depends on the geometry and the Reynolds number.

 

Frequency of a "string" under tension is given by:
fn = n / (2 * L) * Sqrt(Tau/Rho)
where
n is the mode; 1 for first mode, 2 for second, etc
L is the length of the string,
Tau is the nominal stress, tension divided by cross sectional area
Rho is the nominal density, mass divided by (length X cross sectional area)

 

For the keel and rudder the frequency should be calculated for the trailing edge thickness at least for the zero angle situation.

f=St*V/L

For a normal 5-10 mm trailing edge at 3-6 m/s speed and with "normal" Strouhal number of ~0.2 the frequncy would be 60-240 Hz.

The shrouds usually start at around 10 m/s wind, which would mean ~300 Hz for typical shrouds. The first harmonic is around 20 Hz for a typical shroud, thus n is around 15. I guess there is not enough energy at lower wind speeds although the frequencies would match.

Has anybody tried the frequency method for measuring shroud tension? I have used it for bicycle spokes and engine timing belt tension measurement, but not yet for rigs. It is very accurate for those two, but the frequency shroud harmonic frequency may be too low.

 

Should mention that the formulas I gave for string frequencies neglect rotary inertia which for shrouds should be negligable except for high number models, and bending stiffness which should be negligable for wire and rope rigging. For higher number modes and rod rigging bending stiffness may become significant enough that it would have a measurable effect.