THE DAMPING OF HEAVE AND PITCH: A COMPARISON OF TWO-DIMENSIONAL AND 
THREE-DIMENSIONAL CALCULATIONS 
By PROFESSOR Sik THOMAS H. HAVELOCK, M.A., D.Sc., F.R.S. (Honorary Member and Associate Member of Council) 
1. Damping coefficients for heave and pitch are usually derived 
by calculating the mean rate at which energy travels outwards 
in the wave motion produced by the oscillations. The calcula- 
tion is based upon approximate solutions for the two-dimensional 
motion due to heaving oscillations of a long cylindrical floating 
solid; the application to heaving and pitching for a ship then 
proceeds by the so-called strip method. Each thin section of the 
ship is treated as part of an infinite cylinder of the cross- 
section at that point, sending out two-dimensional waves on 
either side. The coefficients for the ship are obtained by inte- 
grating along the length of the ship. Reference may be made to 
Weinblum and St. Denis(2) for a detailed exposition with cal- 
culations. In the work of those authors no allowance was made 
~for the difference between the assumed flow and the actual 
three-dimensional flow; this may be justified to some extent in 
that results in practical cases seem to give reasonable agree- 
ment for heaving, but the application to pitching requires more 
consideration. 
In discussing this point, Korvin-Kroukovsky and Lewis®) 
remark that the damping coefficient for heaving may be assumed 
tc be correctly represented by the two-dimensional calculation, 
but they adopt an empirical reduction factor of one-half for the 
similar calculations for pitching. 
In a recent paper Korvin-Kroukovksy™) discusses the matter 
in considerable detail, and expresses the opinion that an important 
effect of three-dimensional flow may exist. He estimated the 
validity of the two-dimensional calculations by comparing the 
data with results from towing-tank experiments on two models. 
It was found that, at the natural frequencies of the models, the 
results were in substantial agreement both for heaving and for 
pitching within the limits of experimental error, which were 
admittedly rather wide limits. However, for more extended 
ranges of frequencies, it was found necessary to introduce 
empirical correction factors, in one case, for instance, reducing 
the damping coefficient for pitching to 75 per cent of the cal- 
culated value. Korvin-Kroukovsky remarks: “In the case of 
damping in heave, most of the force comes from the middle part 
of the body where the flow hardly differs from the assumed two- 
dimensional one. The good agreement in regard to damping in 
heave was therefore not surprising. The close agreement in the 
damping in pitch was not expected, however, and in fact was later 
not confirmed in the application of the calculations to the entire 
set of model motions. Most of the contribution to the moment 
coefficient comes from the ends of the ship, where one logically 
_ Should expect a large change from the assumed two-dimensional 
flow to the actual three-dimensional flow.” It is clear that the 
matter is not in a very satisfactory state, especially as the use of 
an inclusive empirical factor may hinder recognition of the true 
cause of the discrepancy. 
2. The present work is intended, not as a solution of the 
problem, but as a contribution towards elucidating the particular 
point of the difference between two- and three-dimensional 
calculations. Of course the only really satisfactory method would 
be to work out the problem for a floating solid. It is not difficult 
to formulate the mathematical equations; but even for a simple 
form, such as a spheroid half immersed, the expressions soon 
become very complicated and numerical computation of pro- 
hibitive length. In this paper we deal with the simpler problem 
of a solid which is wholly immersed in the water, and we obtain 
the damping coefficients by the two methods: strip-method and 
three-dimensional. Although the separate results would not be 
applicable to a surface ship, it is thought that the ratios of the 
coefficients obtained by the two methods should at least give a 
useful indication of the sort of difference that might be expected. 
The calculations are given in the Appendix, comprising the basic 
theory, application to a submerged spheroid, approximate 
expressions for any elongated solid of revolution, and some 
remarks on the general ellipsoid with unequal axes. 
3. We consider now some numerical results. for a spheroid 
submerged in water with its axis horizontal. The spheroid is 
made to describe (i) heaving oscillations, (ii) pitching oscillations. 
Ey is the rate of energy loss for heaving calculated from three- 
dimensional flow, Eys from the strip method. The corresponding 
damping coefficients in the equations of motion of the solid are 
directly proportional to the energy loss; thus E,;/Eys is the ratio 
of the coefficients by the two methods. Similarly, for pitching 
Ep/Eps is the required ratio. The general formulae are given 
in (23) and (24). We take a spheroid with a length-beam 
ratio of 8, as a fair value for comparison with ship models; in 
this case e = 0-996, ky = 0-945, k’ = 0-84. With these values 
(23) and (24) were computed for integral values of kg a, that is 
of o2a/g, up to 10. The results are shown in Fig. 1 on a base 
fe) 6 8 10 12 14 lb 18 20 
oly 
Fic. 1.—RATIOS OF DAMPING COEFFICIENTS FOR HEAVING AND FOR 
PITCHING 
of o? Lig, where o is the circular frequency of the oscillations 
and L the length of the model; this seems to be the suitable 
parameter for comparison with model results. 
The ratio for heaving rises rapidly at first and attains a maxi- 
mum of about 1-1, and then, with small alternations, it soon 
approaches unity. With the sort of accuracy attainable prac- 
tically, the ratio may be taken as unity when o? L/g is greater 
than about 6. The ratio for pitching rises slowly at first and then 
very rapidly up to a maximum of about 1-15. In this case it 
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