I am looking for papers/books presenting the regression equation between body size (body mass or wing length) and wingbeat frequency in hummingbirds. I'm aware of the two excellent publications below, and I try to get a hand on some of the old publications of Greenewalt, C. H.. Is there other publications that are addressing this relationship between body size and wingbeat frequencies in hummingbirds? I will need the regression equation to predict wingbeat frequency as a function of body size.
Altshuler, D. L., and Dudley, R. (2003). Kinematics of hovering hummingbird flight along simulated and natural elevational gradients. Journal of Experimental Biology 206, 3139-3147. doi:10.1242/jeb.00540
Altshuler, D. L., Dudley, R., Heredia, S. M., and McGuire, J. A. (2010). Allometry of hummingbird lifting performance. Journal of Experimental Biology 213, 725-734. doi:10.1242/jeb.037002
Although not specifically for hummingbirds Colin Pennycuick came up with an equation to predict wingbeat frequency: Pennycuick CJ. 1990. Predicting wingbeat frequency and wavelength of birds. Journal of Experimental Biology. 150: 171-185.
Again, not specific for hummingbirds, but a complementary work to Pennycuick's work is: Bruderer, B; Peter, D; Boldt, A; Liechti, F. 2010. "Wing‐beat characteristics of birds recorded with tracking radar and cine camera", Ibis 152: 272-291.
This one might bring some info to what you need.
Robert L. Nudds, Graham K. Taylor and Adrian L. R. 2004. Thomas of Strouhal number for high propulsive efficiency accurately predicts how wingbeat frequency and stroke amplitude relate and scale with size and flight speed in birds.Proc. R. Soc. Lond. B. 271: 2071–2076.
doi:10.1098/rspb.2004.2838
Good luck, and let us know when you find it.
Why do you not use the equation (wingbeat frequence [Hz]=39.3–1.6xbody mass [g]) as given in Altshuler & Dudley (2003)?
This should work...
Thanks for replies. I will search for the references when I'm back at the office tomorrow. To Heiko, I will use the references to compare our own results. Some students and me have filmed free-ranging hummingbirds in Costa Rica, and I would like to compare if our equation gives similar predictions as others.
I cannot refer to any paper, but I could measure the sound of the hum for quite a few species and compare it to the body size.
Niels Krabbe,
I might be mistaken, but wouldn't the hum be more influenced by the wing size, shape and area, beat frequency and weight, than to the body size?
I am inclined to think that these parameters are not entirely correlated.
I would suspect them to be entirely correlated. The hum is the beat frequency.
I'm aware of one study using sound recording equipment:
Dong, J. P., F. C. Governali, E. I. Larson, S. A. Snow, and E. V. A. Unger. 2010. Wingbeat frequency is related to foraging strategies of hummingbirds at Monteverde, Costa Rica. Dartmouth Studies in Tropical Biology 19:26-29.
They don't present any equations, only wingbeat frequencies for each of the hummingbird recorded.
Hi Ronny,
In page 89 of the book "The biomechanics of insect flight / Robert Dudley 2002" there is a figure with the relationship between log body mass and log wingbeat frequency. In the legend it is stated that "wingbeat frequency of hummingbirds scales with mass ^-0.61".
Cheers
Nir
I have only just seen the question, so apologies for the late response which I hope is still of some use. A long time ago I did the analysis based on the (then) available data, mainly from Greenewalt. The formulae are in the attached papers (I think one or other of these was the source for Robert Dudley's formula). There are definitely more numbers available now, but I doubt if adding them would change the scaling model very much. Note that the maximum frequencies are around 80 Hz: I never found any substantiated records of frequencies up to 200 Hz as claimed by some authors, but I have no first hand evidence either way.
There are a few things I'd advise you to note. First, the scaling for hummingbirds is very different from that for other birds, no doubt because of their small size, hovering flight and (probably most important) distinct wing morphology and wingbeat kinematics. Second, the scaling will give you an approximate indication of the frequency, but _only_ an approximation. At the lightest size range of hummingbirds (2-3g) the hovering frequency can vary between around 35 and 80Hz, depending on the species, while the scaling equation would predict around 50Hz. In principle better estimate could be made by adding the wingspan and wing area to a multiple regression formula, but the dataset I have is much too small to compute this with any accuracy. Third, according to the dataset (and sometimes individual sources such as Greenewalt) frequency seems to be highly variable within a species, and might vary for instance with gender, size/mass or relative wing size (or some of the measurements may be inaccurate). Other factors will be involved as well, including altitude, environmental conditions, body condition, whether recently fed, and so on. I am not sure there are enough data out there to look seriously at the problem - there certainly were not when I was looking at this - but it would be an interesting study.
Finally - as I hope the papers explain - there is no one single biomechanical factor that determines frequency as a function of size. Rather, the perceived scaling (allometry) is a summary of the evolution of wing design and frequency within a variety of constraints, which will have different significance on birds with different habits. So I have always been cautious about reading too much biological meaning, or indication about the biology of an individual species, into a scaling formula.
I hope this is of use to you. If you still need copies of the Greenewalt data, or relevant bits of his book and papers, let me know.
Regards
Jeremy Rayner
Article Form and Function in Avian Flight
Article Flight mechanics and constraints on flight performance
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