PSI - Issue 20

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

291

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

8

We get the first approximation for hydrodynamic forces acting on the profiles. Using expressions for the potential (35) – (38) and for forces proportional to the acceleration from (41) we obtain:

     

     

     

     







  k i   k i

2 F a V     ix

2

V

V

2

cos 2

sin 2

,











(44)

kx

ky

ix

ik

ik

ik

1







2 F a V     iy

2

V

V

2

cos 2

sin 2

.











(45)

kx

ky

iy

ik

ik

ik

1

  ˆ 2 2 M N N  . Elements of this matrix

Let's define connection matrix under acceleration

x y ' can be found from (44), (45). These values have a simple physical implication, for example, i k 

 '

   

   

i k

x x





i k

i k

y x

y y

x y i k '  is the opposite sign force acting on the i- th profile in the OX direction due to the movement of k- th profile in the OY direction with a unit acceleration. Taking into account matrices coefficients, these relations will look like: ' 0, ' 1,     y x x y x x x x i i i i i i i i     2 sin 2 , cos 2 , ' 2 2 2 ik ik i k ik ik i k x y x x           (46) . cos2 , if 2 sin 2 , ' 2 ' 2 2 i k ik ik i k ik ik i k x y y x            i.e., acceleration connection matrix is symmetric. We calculate the first approximation for forces proportional to the oscillating profiles velocities. Let's use expressions (35) – (38), (40), (42). In the expression (42), when calculating the time derivative from i Ф _ , we use the relations:           , cos sin 1 , sin cos ik iy ky ik ix kx ik ik ik iy ky ik ix kx ik V V V V dt R d V V V V dt dR               (47) then obtain:         . cos3 sin 3 8 sin 3 cos3 8 3 1 0 3 1 0     ik ky ik kx ij iy k i ik ky ik kx ij ix V a V V S V a V V S           (48) As it is obvious that ki ik    and      ki ik (fig. 1), so , 1 ,...,N , i,k б б , в   в , в ' б ' y y y y x x x x x y x y i k i k i k i k i k i k 

    k i

 '

 '

   

   

 ˆ 2 2 V N N  can be defined from (48). Matrix elements

Connections velocity matrix 

i k

i k

x x

x y

are the





i k

i k

y x

y y