Pien and Strom-Tejsen 



The four surface integrations of Eqs. (68) are all similar. Hence, the in- 

 duced fluid velocity for the case of periodic loading is found in just the same way 

 as for the steady case, except that more computational work is involved. 



Equations (67) through (72) constitute the main body of our results. They 

 relate the induced absolute fluid velocity components to the blade loading, which 

 can be steady or periodic. Even though these equations are derived only for 

 time t being equal to zero at a fixed space, they actually relate the induced 

 absolute fluid velocity component to the blade loading at all times in the space 

 relative to the blade position. This is because whatever is valid in the steady 

 case as observed on the blade for t being equal to zero is valid for t being 

 equal to any value. It is also true for periodic blade loading. The amplitude of a 

 sinusoidal function determined from the real and the imaginary values at any 

 time is the same. 



Within our scheme of evaluating kernel functions and representing the load- 

 ing function components the surface integration involved in Eqs. (68), (70), and 

 (72) can be conveniently carried out since the integration in the chordwise direc- 

 tion can also be performed functionally. The next section gives the details of 

 such surface integration. 



Integration Over the Lifting Surface '- ' 



The surface integrals as discussed in the previous section are all alike. The 

 integrand of each of the surface integrals involves a product of a loading function 

 and an appropriate kernel function. Since each of the kernel functions is a linear 

 combination of 1 3", 1 5% and l°, and since the loading function is expressed as a 

 polynomial of 6? - in the chordwise direction, the results of the chordwise inte- 

 gration of the product of a kernel function and a loading function is a linear com- 

 bination of the quantities 



j?og = f (d-4>r log 



where n = 0, 1, 2, ... , 



^^ "'^^ + ^/^(V^^^Tf~~^^2h(e^^^)~+~~^ 



d0 , (73a) 



J^n, __ I ' (0-^r dg , ' (73b) 



^L y/a(e- 0)2 + 2b(^- 0) + c 



where m = 0, 1, 2, . . . , and 



J3'" = I' (i^^r_de (73^) 



^L [a(0-0)2 + 2h(0-4>) + cf^^ 



where m = 0, 1,2,3, .... 



110 



