PSI - Issue 20

Tatiana Fesenko et al. / Procedia Structural Integrity 20 (2019) 284–293 Tatiana Fesenko et al. / Structural Integrity Procedia 00 (2019) 000–000

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coefficients with opposite sign at the corresponding velocities in (48). For example, x y i k '  is the force, taken with the opposite sign, acting on the i- th profile in the OX direction due to the movement of the k- th profile in the OY direction with a unit velocity. From (48), taking into consideration matrices coefficients, we obtain:

' '  

4             cos3 , ik 4 ' 3 3 i i ik i i i i i i x y y x y y 

0

i i

x x

4 sin 3 , 3 

4 sin 3 , 3  

'  

 

i i   y x

 

(49)

i i

ik

ik

ik

ik

x x

x y

i k 

when cos3 ,

.



i i

ik

ik

y y

You can verify that the computed matrix C is antisymmetric, i.e.

.  Positional force in the first approximation can be calculated using expressions (35), (36), (40) and (43): sin3 . 4 cos3 , 4 1 3 2 1 0 1 3 2 1 0         k ik ik iy k ik ik ix aV P aV P       (50) We determine the force increment due to a small configuration change. Let j- th profile ( j = 1,..., N ) moved along OX to xj and along OY on yj , and xj and yj are significantly less distances between tubes, then:           , cos sin 1 sin cos ik i k ik i k ik ik ik i k ik i k ik y y x x R y y R x x                (51) where ik R  and ik  – small changes of ik R and ik  due to small displacements of i- th and k- th profiles. We obtain:             , cos 4 sin 4 12 sin 4 cos 4 12 1 3 2 1 0 1 3 2 1 0             k ik i k ik i k ik iy k ik i k ik i k ik ix y y x x V P y y x x V P           (52) iy P  is a change of positional force corresponding components. The matrix of positional communication K can be determined by analogy with the determination of matrices C and , , 1,..., ' , x x k i '  , ' i k N y y y y x x x y y x k i i k i k k i i k            where ix P  and

  '

 '

   

   

i k

i k

, found from (43), have the following form taking into account matrices

x x

x y

M. Matrix K elements



i k

i k

y x

y y

coefficients:

j i  

4

4  ij

' 

12

cos 4 ,

'    i k x x i k x x

12

cos 4

    i i y y

   







i i

ij

ij

ij

x x

(53)

j i  

4

4  ij

' 

12

sin 4 ,

'    i k x y

12

sin 4

.  

    i i y x





   

i i

ij

ij

i k

ij

x y

y x

To study elastic tube bundle small oscillations, parameters value ε �� and γ �� , included in the expression for matrix