PSI - Issue 20

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

285

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

2

vibration amplitudes and their destruction as follows from Fesenko and Fursov (2005):

  

  

  

  

  





 

     

     

M C K S S

    S F

.

(1)

Matrices [M], [C], [K] have dimension k k 2 2  , where k – tubes number. In fluid flow, the matrices [C], [K] are not symmetric.

Nomenclature a

streamlined profile radius, m

[C] damping matric F ix , F iy force components proportional to accelerations of the oscillating profiles on the OX and OY axes, N [K] stiffness matric [M] mass matric p fluid pressure, Pa P ix , P iy force components proportional to profile displacements on OX and OY axes, N r i , θ i fixed polar coordinates, m, rad S ix , S iy force components proportional to the velocities of the profiles in the OX and OY axes, N V 0 approach flow velocity from infinity, m/s

V ix , V iy components of i- th oscillating profile instantaneous velocity m/s α i,j , β i,j coefficients of hydrodynamic communication for tubes acceleration α i,j , β i,j coefficients of hydrodynamic communication for tubes velocity γ ij , R ij j -th profile polar coordinates in the i -th coordinate system, rad, m Δ Laplace operator ε small parameter ρ 1 liquid density, kg/m 3 v i,j , µ i,j coefficients of hydrodynamic communication for tubes displacement Ф flow potential

To determine the dynamic response of the tube arrays it is necessary to solve numerically 2 k systems of differential equations with respect to generalized coordinates (1). For solution of the dynamic problem (1) step-by step Wilson integration method has been used as this method can be used for arbitrary structure of elements with physical and geometric nonlinearities. The proposed vibrations of tube bundles in cross flow mathematical model allows us to study its frequency properties and show that at a flow velocity greater than a certain critical value, tubes bending deformations substantially increase. In order to realize the mathematical vibrations model (1) it is necessary to determine the influence matrix in equation (1), which will determine the critical flow velocity for specific tube bundles, to do this, we determine the coefficients of hydrodynamic communications. We assume that small oscillations of tube bundles take place in the linear region consisting of k identical elastic tubes in cross flow. In the non-deformable state, the axes of all tubes are parallel to each other, tube length ( L ) considerably greater than tube radius ( a ), and distance between tubes and oscillation of each tube can occur on one of the first bending oscillations forms with a characteristic scale of L order along the z axis parallel to tubes axes. This assumption gives the basis for distributed hydrodynamic forces calculation in each section z = const to consider flow as plane, i.e. to consider the variable z as a parameter. Also assume the fluid is ideal and incompressible and flow around each profile (at z = const) irrotational. This makes it possible to determine the parameters of hydroelastic excitation without the influence of other types of excitation We determine hydrodynamic relationship that occurs when a potential non-circular flow around the system of identical circular profiles, performing small oscillations. In this case, the hydrodynamic force acting on the profile, we call the distributed force acting in z = const section per tube length unit from liquid. Obtained expressions for