Smith and Cummins 

 where 



tan e^j = R- j(^)/Ri j(a;) , 



Rij(^) = ^ii^^"^^ ^°^ '^'^ '^'^ ' 



R..(aj) = I ^ij('T) sin oiT dr . 

 



We will make use of the complex function 



R*j(^) = rIj(«) + iRlj(^) • 



The frequency dependent functions, R^j and Rf j , are the in-phase and out-of- 

 phase responses in the jth mode to a sinusoidal excitation in the ith mode. 



The conventional manner of writing the system of coupled equations of mo- 

 tion is 



6 



E (^3jk^"j + bj^x. + c.^x.) = Fk cos(a)t+ek) , k=l,2,...,6. (3) 



j = i 



Using Eqs. (2) and (3), it is possible to develop a system of equations between 

 the coefficients a^j, b-^ and the functions R^^ R?j. The Cj^ are assumed to 

 be determined from static tests. Thus, from the matrix [Rij(cc))] we can deter- 

 mine the matrices [a. 1 and [b. -1. 



Experimentally, enough information can be obtained from a series of six 

 tests in which the six sets of excitations are linearly independent to determine 

 [r^j] and subsequently the coefficients. 



As we stated above, we have restricted ourselves to the two modes of heave 

 and pitch. The model was free to respond in all six modes, and all restraints 

 were measured, but in the analysis it was assumed that the coupling between 

 these two modes and the remaining modes was negligible. 



Experiment I 



The model was excited in heave and pitch by imposing a pulse f(t) at the 

 bow (see Fig. 1). The excitations were i^Ct) = f(t) +g(t) and fgCt) = --1 • f(t), 

 where ^ is the distance of application of f(t) from the center of gravity and 

 g(t) is any constraining force in heave. (There is no constraining moment in 

 pitch.) The responses were ^^(t) and x^ct) . As discussed above, the excita- 

 tions and responses were resolved into Fourier series: 



450 



