Ogilvie 



bending moment the hypothesis of superposability proves somewhat poorer than 

 for pitch and heave motions. At F = 0.18 there is already considerable discrep- 

 ancy in bending moment f.r. functions computed from spectra in different sea 

 states, and coherency is fovind to run much lower in general. 



Gerritsma (1960) arrived at similar conclusions in tests with Series 60 

 models. His approach was somewhat different from Dalzell's. Gerritsma found 

 the f.r. functions in heave and pitch by three different methods: (1) a direct de- 

 termination from measured ship responses in small amplitude regular waves; 

 (2) calculation from the second order ordinary differential equations of motion, 

 after an experimental determination of coefficients in the equations; (3) tests in 

 irregular waves, in the manner of Dalzell. 



Most of Gerritsma 's regular -wave experiments were performed with a 

 wave amplitude 1/48 of the model length, L. He also tried larger amplitude 

 waves, 1/40 L and 1/30 L, for the Cg = 0.70 and Cg = 0.80 models. There was 

 generally excellent agreement among responses to the waves of various ampli- 

 tudes, but a reduction in response appeared in some cases for the 1/30L wave. 

 There was no pattern to the lack of linearity which could be associated sys- 

 tematically with either speed or wavelength. (Froude number was varied be- 

 tween 0.15 and 0.30, VL between 0.75 and 1.75.) 



These regular waves were much less steep and much smaller in amplitude 

 than the severe irregular waves used by Dalzell. It appears that nonlinear ities 

 make themselves felt more easily in regular waves than in irregular waves. 

 This has also been observed recently by Ochi (1964),* who showed that f.r. func- 

 tions must be obtained from small amplitude waves, if regular waves are to be 

 used at all for their determination. These f.r. functions can then be used in 

 confused seas of much greater severity. One is tempted to argue that in a con- 

 fused sea the amplitude of a wave of any particvilar frequency is infinitesimally 

 small and so the f.r. functions for infinitesimal amplitudes should be used — 

 even though the actual wave heights and steepnesses may be very large. Of 

 course, such an argument is illogical and explains nothing. For the moment, we 

 must simply accept the phenomenological observations described above. 



Gerritsma also performed a series of tests in which he determined the 

 coefficients (i.e., added mass, added moment of inertia, damping, buoyancy, and 

 couplings) in the differential equations of motion for heave and pitch. This was 

 done by forcing the model to oscillate in various ways in calm water. Then the 

 model was restrained and towed in regular waves, measurements being made of 

 the heave force and pitch moment. With all of these quantities known, he solved 

 the equations to obtain the f.r. functions. 



*Ochi has also pointed out that acceleration nrieasurements are much more sen- 

 sitive to nonlinearities than are displacement measurements. This is simple 

 to explain. In regular waves with frequency of encounter 6j, we could represent 

 the effects of nonlinearities by expressing the vertical displacement of a ship 

 reference point by a Fourier series, the terms being sinusoidal with frequency 



no), n = 1, 2 If acceleration can be expressed by differentiating this series 



twice, then the terms in the Fourier series for acceleration will be multiplied 

 respectively by n^, so that the higher order terms are relatively larger than in 

 the displacement Fourier series. 



16 



