Theory of Unsteady Propeller Forces 



velocities on S^ and Sr are periodic functions of period 27t/n with respect to 

 time s and that gi(^,v,-S^) and the induced velocity on Sp are periodic functions 

 of period 2tt. Thus the quantities gi(^,v, -5^), m*{l.^,r]^,s), and l^(ri^, u,s) can 

 be expanded formally as 



n m 



m " , , . 



where 



- imNs 



(91) 



gRmC^r") = gRmo(^l)/r^+ gRml(^l) ^1 " "' + gRm2(^l) " Vl " U^ + • 



gR(^l-")= gR0(^l)/fT^+ gRi(^l) Vl - U^ + g^2(^l) U Vl - U^ + 



' (92) 



and g„o(^), gRn,o(^i)» g^o(^i)' etc., are functions of ^ or Vr From Eqs. (70) 

 we get the relations 



g^o(^l) = 4°(^l) • Z SR-o(^i) e-^"^^ = a*(r,^,s) 



(93) 



We define the following functions: 



.- imNs 



Bz* R* 



= H^ , H e- 



imNs _ „- imNs 



J(SH) 



d 1 



II „- imNs 

 "k 'yHK ^ 



J(SH) 



-H,,,„^ e' 



imNs 



J(SH) 



B r imN[q)/Vjj(i7j)-sJ d 1 





Bz* R = 



nN(9/i/^(T7 J )-s) B 1 



= R 



LxH K m 



J (SH) 



Bt] J„ 



R 



LyHK m 



J (SH) 



_1_ f" imN[<p/v (7,i)-s] _^ J_ 



Bz* ^ Bz* R* ^'P 



Jq ^ (SH) 



45 



= R, 



. - i mN s 



■ imNs 



(94) 

 (Cont) 



