Smith and Cummins 



Fourier transform of the rectangular pulse will be zero for the frequencies Aw, 

 2Z\w, 3Aw, . , . where Aw = i-n/T . At these frequencies, the transform of f'^(t) 

 will have the same value as the transform of f ^Ct). Therefore, if we restrict 

 our calculations to this set of frequencies, we need not concern ourselves with 

 estimating f^ . 



Another way of treating the same problem is to consider i'^(t) to be one 

 cycle of a periodic signal. This periodic signal would be the response to a pe- 

 riodic sequence of pulses with period T. Such an analysis is permissible be- 

 cause the memory of model is less than t, and the effect of all previous pulses 

 will have dissipated before the start of a new pulse in the sequence. This peri- 

 odic signal can be analyzed in a Fourier series. The effect of fg will appear in 

 the constant term but in none of the others. Thus, the Fourier coefficients cor- 

 responding to the frequencies Aw, 2Aaj, 3Aw, . . , completely define the function, 

 where Aw has the same meaning as before. But these Fourier series coefficients 

 are identical with the values of the Fourier integral transforms of f^(t) and 

 f i(t) at the same set of frequencies, and we arrive at the same conclusion as 

 in the analysis of a sir^le pulse. 



This periodic t3^e analysis was used for all test data in order to eliminate 

 any dc components. The basic assumption, which is inherent in any truncation 

 process, is that the transient has completely decayed before the instant of trun- 

 cation. This assumption will never be strictly correct, and the degree to which 

 it is not fulfilled may be an important source of error. 



It should be noted that the phases obtained from this periodic type analysis 

 are referred to the arbitrary starting instant and are therefore meaningless in 

 terms of the physical test. If, however, only the phase difference between data 

 channels are considered, the results immediately become physically meaningful. 



The data analysis sequence is as shown in Fig. 7. Suitable computer pro- 

 grams were written to analyze the data, to invert the resulting matrix of coeffi- 

 cients (see following section), and to compute the damping and added mass terms 

 for each of the harmonics considered. Also, a computer was programmed to 

 plot the resulting a's and b's versus the nondimensional frequency, w^/L/g. 



Equations of Motion 



The relation between the excitations and responses of a ship, under the as- 

 sumption of linearity, can be written in various ways. In terms of the impulse 

 response functions [1] we have 



6 CO 



xj(t) = ^J Ri.(T) f.(t-T) dr j = 1,2,...,6 (1) 



i = l 



where Rij(t) is the response in the jth mode to a unit impulse at t = in the 

 ith mode. The matrix of functions R. (t) thus completely characterizes the 

 response of a ship to an arbitrary set of excitations. In the following discussion 

 we adopt the convention: 



448 



