non-linearities, we have to be careful to explain precisely what 

 is meant by this distinction (linear versus non-linear) which is 

 illustrated by the second level in the hierarchy of model 

 characteristics (Figure 1). 



In summary, if the system is strictly linear, the output 

 statistics can be obtained directly. If the system in non-linear, 

 an exact analytical derivation (of the output statistics) is 

 generally impossible, except in very specialized cases (Caughey, 

 1963). Thus, two alternatives are possible: an approximate 

 analytical solution, or an experimental determination of the 

 response statistics by means of analog or digital simulations. 



Approximate Analytical Techniques . If the systemj inputs 

 are small, the usual approach, as cited above, is to linearize 

 the non-linear equations and derive the output statistics from 

 the resultant approximate linear model. This has been applied 

 to problems of vibrating strings (Caughey, 1963) and ship rolling 

 (Haddara, 1973) for the purpose of deriving response moments of 

 first and second order. If the system non-linearities are small, 

 then first and second order response moments for large inputs 

 may often be obtained by using a classical perturbation approach 

 (Crandall, 1963). For the moored ship, the first and second 

 moments are important but they are not the only important statis- 

 tics — estimates of, for example, peak distributions and the 

 probabilities of exceeding maximum allowable values of stresses 

 or displacements, especially under heavy sea conditions, are 

 also needed. 



Simulation Techniques. In order to obtain satisfactory 

 estimates of system responses by simulation, long duration rurls 

 are required. This means that simulation algorithms used should 

 be efficient as well as accurate, with the sampling rate close 

 to the Nyquist rate for the bandlimited signals. It also means 

 that algorithms for generating synthetic test-input sequences 

 on the computer must be capable of being adjusted to give spectral 

 characteristics and amplitude distributions similar to those 



15 



