Response of Propulsors to Turbulence 



00 CO 



(6) 



where ^ll(r) is the velocity correlation tensor for the points located at y^. and 

 Yr respectively. 



The spectrum tensor of the forces acting on the propulsor is the Fourier 

 transform of the force correlation tensor <i>°fi(T). It consists of the sum of the 

 spectral and cross- spectral densities of the forces acting on the individual seg- 

 ments of the propulsor. The spectrum tensor is given by 



G°/5(a.) = :^ r (D^^Cr) e-i"^ dr 



-00 



CD 



= F f E E -^"fe.CO e--- dr 



-"-co i j ' ^ 



(7) 



where i = \f^. It is convenient to express the spectrum tensor in terms of the 

 frequency response functions of the individual segments of the propulsor. From 

 Eq. (6) we obtain 



00 CD CO 



= [ FflTC-i) e'"^' dr, f Ff^Cr,) e'"'^"^ dr^ 1 f R^^^(r) e'-^dr 



-b -^0 - CO 



= [H-^(-)]* K(^')]<(^-) ^ ■"/ ■ (8) 



where the frequency response function H(a)) is given by 



Hf,(.) = ( Ff^Cr,) e"^"^^ dr, 



and where the * denotes the complex conjugate of the quantity. The spectrum 

 tensor of the turbulence has the form 



CO 

 ~0D 



295 



