Testing Ship Models in Transient Waves 



INPUT 



* G(]a.) 



y (t) 



OUTPUT 



Fig. 1 - General linear 

 system representation 



y(t) as the dependent or forced variable. If x (t) 



x(t) is a sinusoidal signal at a particular 

 frequency, then, in general, y(t) will as- 

 ymptotically approach a steady- state sinus- 

 oidal response at the frequency. The ratio 

 of the output amplitude to the input amplitude 

 and the phase difference between output and 

 input for all frequencies define the frequency 



response of this system, represented by the complex transfer function G( jw), 

 where ^ is the frequency in radians per second. 



When the system G(jaj) is at rest and a sudden transient x(t) is applied at 

 t = 0, then some response y(t) will be measured, usually involving decaying 

 transients. It is well known that a transient signal can be decomposed into a 

 continuous distribution of infinitesimal sinusoidal components with the aid of the 

 Fourier transform. For example, the Fourier transform X(ico) of a particular 

 input signal x(t) is given by the complex quantity 



X(jc^) 



f 



dt x(t) e 



j <i) t 



(1) 



which represents the amplitude and phase of the incremental components at fre- 

 quency CO. Considering the output to be a summation of the response to each of 

 the input frequencies, the well-known relation 



Y(}co) = X(ja;) G(ja;) (2) 



gives a proper amplification and phase change to each of these components. 



To summarize, the frequency response of a system G( joj) can be found 

 from a single transient experiment with input and output transforms X( j^j) and 

 Y( jw) , respectively, with the relation 



G(ja;) 



Y(ico) 

 X(ico) 



(3) 



In the transient testing of ship models, the input x(t) is arbitrarily defined 

 as the instantaneous amplitude of the undisturbed two-dimensional wave surface 

 which would pass through the center of gravity of the model; see Fig. 2. The 

 output y(t) is the time history of any one of the pertinent response variables, 

 such as roll, pitch, or heave. 



The use of a water wave input is the key distinction between transient tests 

 of ship models and those conducted, for example, in control systems analysis 

 where often a voltage is available for easy introduction of input transients. 



Visualization of the Ship-Wave System 



The fact that the wave height referenced to the center of gravity of the 

 model is defined as the input can lead to great mathematical difficulties. For 



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