where 



Theory of Unsteady Propeller Forces 



X* = y\e^{^') + ^(^')Vj] - i - v^s , &*= 0^(^')+0(^')v^ 



R* = yx*2 + ^-2 + ^2 _ 2^" if COS 0* , ...:•- .: ■; 



g*i(^'-v', - S^') - ( gi{?",v', - [e(^")(v^-v') + S^,]} dv' , 

 -1 



gj(^',v',-S^,)=0, for v'>l or v'<-l. • •.■ 



From Eqs. (22) and (23) we get 



(103) 



(104) 



The velocity components induced by </)pj in the x, r, and t^ directions, which 

 are denoted by w^,,,^, w*j,^, and wp^^ respectively, are obtained by differentiating 

 partially with respect to C, ^, and 6; then integrating them by parts, referring 

 to Eqs. (104), the following expressions are obtained: t?: - ^ . . ;i ; 



''PSx 



B^ 



1 N CO 



So k = 1 1 



'g*i(^'''Vi' -^k') ^'(^' -^ cos 0*) 



B^' 



(^') 



Bg*i(^'.Vi, -S^,) ,c sin 0* - ^'(^' -^ cos 0*) 30*/B<f' 



3v, 



dv^, 



'PC 



^Ptr 



B^ 



f|-^- L J, 



Bgt(^'--i--Sk') 



3^' 



^' sin 0*BXV^(^')3vj-<f'X* cos 0* _ 3g*(<f',v^, -S^,) 

 -ri ^(^") + ; 



X* sin 0* - ^' sin 0*3X*/B<f' + ^'X* cos 0*B0*/Bs^' 



R^ 



dv^. 



^35 ' 4 



i B k '= 1 1 



^gt(^'-Vi--Sk-) 



(^-^' cos ©*) 3XVf^(^')Bv. - ^'X* sin 0* _ ^g^C^'-v., -S^,) 



i 0(^') + —^ — !^ ^ 



R3 



X* cos 0* - (^- ^' cos ©*) 3X*/3e' + <f' X* sin 0* B0*/B^' 



R*3 



dv. 



(105) 



53 



