Propeller Excitation and Response of 230000 TDW Tankers 



4 ir tf = — — 



U A U A U 



7 tfx aa r ° T * A x ^t a 



— — ArAx+- - Ax Ar +-3—3— Ax 



d dt d dr d dx 



1 dT A A 1 d* A ■ 1 <? r A a 



+ - — Ax Ar + — -— — Ar At +— — — AxAt 

 d dt d at d at 



Ar 



(1) 



where 



d 



At 

 A x 



A r 



t 



U 



t ,x, r 



= distance between volume source and field point 



= thickness of volume 



= length of volume along helix 



= length of volume along radius 



= time 



= velocity of advance of volume source 



■ The first part of the right hand side of the equation is due to 

 the motion of cavities and derivatives of * , x and r with respect 

 to time are zero. The last part is due to volume variation of cavities. 



For unsteady, irrotational flow the fluctuating pressure p 

 field at a field point : 



P 

 sp 



at 



I" 2 



at 



where 



P 

 s 



U 



4 7T p 



sp 



= density of fluid 



= solid boundary factor |_10 J 



= velocity at the field point, negligible at the hull surface 



Assuming U = U =0 and combining eqs. (l) and (2) : 



2 -, 



-!(I)-*Lu +A_(I)ii + i_ ±j 

 dt V ax x at V at d at' 



Ax Ar 



+ U (l) i» + _L 



[at d' at d 





Ar At + 



i x ar 



1 a 



2 -, 



at W at + d~dt z 



Ax At 



+ -Lll ^ x Ar , 2 ^Tdr 2 3x Sr . 



/1 IT vT^ 1 ^ + "3~iTTr Ax +— - — — A: 

 d dt a t d dt dt d at dt 



(3) 



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