Ogilvie 



2T J_ 



Ryy(T) = E[y(t + r) y(t)] = ^^ | y( t + r) y(t) dt 



can be calculated. Accordingly if we adopt the definition of a spectrum in a 

 wide sense as the Fourier transform of the correlation function, we can com- 

 pute the spectrum. Here a trial has been made to show how the non-linear 

 element — here non-linear damping as described by velocity square damping has 

 been considered — affects the computed spectrum, using an approximation 

 method. For example, for rolling, the equation of motion with velocity square 

 damping is 



10 + Ni<?i + N^4>\4>\ + Ki0 = M(cc) eJ'^*. (3.1) 



Now for the purpose of simplicity, all coefficients on the left-hand side of this 

 equation are considered to be constant and do not vary with frequency. Then for 

 input of a general irregular moment, the equation is 



4> + 2a4> + /30|0| + ^^2^ = g(t) . I (3.2) 



As the zero order approximation 0^, the solution of the linear equation 

 without velocity square damping is taken as 



0^ + 2a0^ + co^^'P^ = g(t) . (3.3) 



Then 



^ CO ^00 



Hg(t-T) g(t)dT = hg(T)g(t-r)dT 



-' - CO '^ - 00 



(3.4) 



hg(T), being the unit impulse response function of this linear system to the 

 compulsory moment g. hg(T) is decided from Eq. (3.3) and, of course, is con- 

 nected to the frequency response function H(a)) by a Fourier transformation and 

 its inverse. 



Here Eq. (3.2) is modified and the compulsory force g(t) is assumed to 

 change to {g(t) - /30|0|}. On substitution of 4>^ into this 4>, the 1st order ap- 

 proximation is taken as the solution of the equation 



0j + 2a0j + ^-^2^j = g(t) - /34>j4>J- (3.5) 



The left-hand side of this equation is just the same as that of Eq. (3.3). There- 

 fore using the same response h^(T) , 



p 00 -CD 



<^l(t) = hg(t-M) {g(M) - ^0o(m) • l4(Ax)|}dM = hg(t-M) g(M) d/x 



-' - 03 J - OD 



-/3 hg(t-M) {0o(/^) • \4>o(f^)\} df^ = '^o(t) - 0l(t) , (3.6) 



•^ - CO 



116 



