Ogilvie 



Fe^^^, and the system parameters, A.^, 



the outputs, X^e^^'', and the f.r. functions are explicitly fvinctions of frequency. 



Such an approach is often described as "working in the frequency domain." 

 The basic set of equations above is valid only if all variables depend sinusoidally 

 on time, at a fixed, given frequency. The utility of this approach follows from 

 two further conditions: (1) The frequency domain analysis can be used for non- 

 sinusoidal motions through use of Fourier transform techniques (for transient 

 disturbances of limited duration) or generalized harmonic analysis (for station- 

 ary random disturbances). (2) Some of the coefficients, A^^, can be interpreted 

 physically in such ways that engineering estimates can be made of the importance 

 for motions of some ship parameters. The first of these points has been dis- 

 cussed at some length in the previous chapter. The second will be considered 

 further here. 



For simplicity, let us consider the experiment in which the model is forced 

 to oscillate sinusoidally in heave only, at angular frequency w. We would meas- 

 ure the amplitude of heave, X3, the six force and moment amplitudes, G., and 

 the six relative phases, {d. - §,)• Then we would calculate the six coefficients: 



ice . -i .) 

 Aj3 = (G./X3) e ^ ^ '\ 



Each of these would be generally a complex number, the value depending on oj. 



In particular, let us look at A3 3. It is related to the f.r. function describing 

 the ship; it equals heave force divided by heave response. We do not know at 

 this point the equations of motion of the ship, but it is an elementary problem 

 to write down an ordinary first order differential equation with constant coeffi- 

 cients which would yield the value ( I/A33) for its f.r. function at a particular 

 frequency. In fact, if we set A3 3 = iajb + c, with b and c real, then the eq\xation 



bi3 + CX3 =: f(t) (2) 



yields exactly (I/A33) as its f.r. function. By considering b = b(a;) and c = c(m), 

 one could use this equation as a description of the pure heave motion of the ship. 

 Such an approach is quite objectionable mathematically, for in simply stating the 

 differential equation above we have implied that we have described the system 

 response for any input, f(t), whereas actually Eq. (2) has no meaning vinless 

 f(t) is a sinusoidal function of time. To quote Tick (1959), "Differential equa- 

 tions with frequency-dependent coefficients are very odd objects." The eqviation 

 has significance only inasmuch as it yields the proper frequency response. In 

 other words, it is not a differential equation at all but is simply another way of 

 writing down the frequency domain properties of the system. 



The naval architect would generally raise a different objection to the use of 

 the above equation: The physical problem involves the dynamics of a rigid body, 

 to which Newton's law is applicable, and, since this law relates the forces to the 



♦They are also functions of relative wave heading, unless we consider only head 

 seas. This point will not be repeated every time it comes up. 



22 



