Theory of Unsteady Propeller Forces 



From Eq. (11) the nondimensional components of the induced velocity are ex- 

 pressed by 



w - , w_ = 



^d0 



(27) 



Then defining the nondimensional relative velocities of water to the propeller 

 blade and the rudder as 



V* = v„ + w* + V 



= W + V, 



y y ly 





V* = w* + V* 



(28) 



the quantities ^)^(^), © i^), W^(^,s), ^^(t^^), and v*^(t;j,s) in Eqs. (26) are 



^ux = ^{^VKTra^3,/v, v*, = iJYfzZtv*]^^^^dv, 



2 77 ' ■ ■ ■ 277 



SR) 



du , 



'0 277^ I kS 



•^0 



V* ds , 



;(^.s) - w*(^, -s,):^y^tr;"^, 



(29) 



and the quantity v (^), which represents the nondimensional pitch of helical free 

 vortex of the propeller, is 



'(<?) 



©(^^^s-^ f^^I.VFii 



3(0* + 0|) 

 ^0 ' — TC ' < 



J(SP) 



/ 277 1 



dv/j-f ds rvf^^ 



/ittH I J_ Vl + V 



3(0S + 0|) 



^ + + V 



^d9 



le 



dv 



(SP) 



(30) 



where the symbols [ ] ^ gp) ^^^ i ] ( sr > indicate that the quantities in the brackets 

 at point ii,^,d) on the surface Sp are represented by Eqs. (18) and at point 

 (i,T],z*) on the surface S,^ are represented by Eqs. (19) respectively (9, 10). 

 Further we may set approximately 



25 



