PSI - Issue 20

Tatiana Fesenko et al. / Procedia Structural Integrity 20 (2019) 284–293

289

Tatiana Fesenkoet al. / Structural Integrity Procedia 00 (2019) 000–000

6

To determine the first approximation of hydrodynamic forces, it is enough to know the potential 1 i Ф taking into consideration two first harmonics. Then it is sufficient to use (28) and (29) for k = 1. Applying the recurrent equation (20), we obtain:

.   

_ 0 _ 1       j i i j

  k

 

(30)

i

From (20), (21) it follows that

2

_ 0

0 r V a

. cos 

 

(31)

j

j

j

Applying (28) for k = 1 and take into account the first two harmonics. Then from (31) we obtain:

   

   

2

R r









_ 0  j

ij

2 V r 

r R 

cos

2   

cos 2 3  

,

.

j 



 

(32)

ij

i

ij

i

ij

i

ij

0

ij

_ 1 i  (22) – (24) we obtain the following expression:    1 2 cos 4 2      

Using for definition

   

   

  

  

r a

 . cos 2 3

_ 1  j

  j i

2

V

i r r a

 



 

(33)

ij

j

ij

i

ij

0

2

R

i

i ij

From (30) we have:

   

   

  

  

  

  

2

4









_ 1

  j i

2

 r r a 2 i

V

 r r a i

   i

  

cos

2 1

 cos 2 3 i 



j 



(34)

ij

ij

ij

0

2

R

i

ij

i

To calculate hydrodynamic forces, the potential value is required only for r i = a :

_ 0

V a

i  

, cos 

2 0

(35)

i



  . cos 2 3    





_ 1

  j i

2

0 V a

2

cos

2   

   i



j 

(36)

ij

ij

i

ij

i  ~ , you can get at r

i = a :

Doing the same, for

~ 0

ix V a

iy V a

i   

 

, sin 

cos

(37)

i

i













~ 1

 j i a V 

2

2

cos

2   

cos 2 3 

   i



j 



  

ij ix

ij

ij

i

ij

(38)



  . sin 2 3    





 j i a V 

2

2

sin

2   





j 



ij iy

ij

ij

i

ij