VIBRATIONS OF A LONGITUDINALLY STIFFENED, LIQUID-FILLED CYLINDRICAL SHELL IN LIQUID

In the paper we study free vibrations of a longitudinally stiffened, viscous liquidfilled orthotropic cylindrical shell in ideal liquid. The Navier – Stokеs linearized equation is used to describe the motion of the internal viscous liquid, the motion of the external liquid is described by a wave equation written in the potential by perturbed velocity. Frequency equation of a longitudinally stiffened orthotropic, viscous liquid-contacting cylindrical shell is obtained on the basis of the Hamilton – Ostrogradsky principle of stationarity of action. Characteristic curves of dependence are constructed.

Introduction. In [1], free vibration of an orthotropic, soil-contacting cylindrical shell inhomogeneous in thickness and stiffened with annular ribs, is studied. Using the Hamilton -Ostrogradsky variational principle, a system of equations of motion of a soil-contacting orthotropic cylindrical shell inhomogeneous in thickness and stiffened with annular ribs, is constructed. To account for heterogeneity of the shell material in thickness it is accepted that the Young modulus and the shell material density are the functions of normal coordinate. Using the Hamilton -Ostrogradsky variational principle the frequency equations are structured and implemented numerically. In the calculation process, linear and parabolic laws are accepted for the heterogeneity function. The characteristic curves of dependence are constructed.
The paper [5] was devoted to investigation of one of the dynamical strength characteristics, the frequency of natural vibrations of an inhomogeneous orthotropic, flowing liquid-filled cylindrical shell made of a fiberglass and stiffened with annular ribs under Navier conditions. The results of calculations of natural frequency of vibrations were represented in the form of dependence of the speed of flowing liquid on the amount of stiffening elements for different values of wave formation parameters and different ratios of elasticity module.
The paper [6] represents the results of finding the frequencies of free vibrations of a structurally anisotropic flowing liquid-filled cylindrical shell made of a fiberglass and stiffened with annular ribs under Navier boundary conditions. The results of calculations of natural frequencies of vibrations are given in the form of dependences on the winging angle of the fiberglass for a shell made of a tissue fiberglass and on the speed of flowing liquid for 1 different values of wave formation parameters and various ratios between the parameters characterizing geometrical sizes of the shell.
The supports formed by the combination of cylindrical panels are used in bridge construction [4]. To save the material, the interior area of the support is filled with soil. Such supports are exposed to different nature forces. One of such forces is a force generated on the surface of cylindrical panels that form supports during flood flow. Under the action of these forces the support is exposed to forced vibration. Therefore, to study the supports formed from combination of cylindrical panels with regard to viscosity and heterogeneity of soil, orthotropic character of panels is of great practical importance. In the paper, based on the Hamilton -Ostrogradsky variational principle, we study forced vibrations of a vertical retaining wall consisting of three orthotropic cylindrical panels contacting with viscous-elastic, heterogeneous soil, obtain analytic expressions to calculate the displacements of the points of cylindrical panels and structure characteristically curves. Account of heterogeneity of soil is performed by accepting its rigidity coefficients as a function of coordinate. It is assumed that the Poisson ratio is constant.
In the paper [9] natural vibrations frequency of the system that consisting of a solid medium-filled elastic-plastic orthotropic cylindrical shell strengthened with discretely distributed rings established on a plane perpendicular to its axis are studied. Utilizing the Hamilton -Ostrogradsky principle, a frequency equation for determining vibration frequencies of the system the following consideration was created; its roots were obtained by mathematical method.
In the paper [3] free vibrations of an orthotropic, laterally stiffened, ideal fluid-filled cylindrical shell inhomogeneous in thickness and in circumferential direction is studied. Using the Hamilton -Ostrogrdasky variational principle, the systems of equations of the motion of an orthotropic, ideal fluid filled cylindrical shell stiffened in thickness and circumference, are constructed.
Problem definition. We consider free vibrations of a longitudinally stiffened viscous liquid-filled cylindrical shell in infinite ideal liquid (Fig. 1). The equation of motion of a longitudinally stiffened orthotropic, liquidfilled cylindrical shell in liquid, is obtained on the basis of Hamilton -Ostrogradsky principle of stationarity of action where 0 A is the total energy of the cylindrical shell, n A is the total energy of the ith longitudinal bar, 1 k is the moment of longitudinal ribs, m A and j A are potential energies of external surface loads acting as viewed from ideal and viscous liquids and applied to the shell and are determined as a work performed by these loads when taking the system from the deformed state to the initial undeformed one and is represented in the form: To describe the motion of external viscous liquid we use the Navier -Stokes linearized equation for viscous compressible liquid [10]: is a vector of velocity of an arbitrary point of liquid, p is pressure at arbitrary point of liquid,  m is liquid density.
The expression of the total energy of the system (2), the equation of motion of ideal liquid (7) and viscous liquid (8) are supplemented by contact conditions. On the contact surface of a shell-ideal liquid we observe continuity of radial velocities and pressures. The condition of impermeability or smoothness of flow at the shell wall is of the form [10] .
Equality of radial pressures as viewed from liquid on the shell: On the contact surface of a shell-viscous liquid continuity of radial velocities and pressures is observed, i.e. for  r R there will be * , , where ,    rx r are viscous forces [1]. It is considered that the rigid contact conditions between the shell and bars are satisfied: We represent the solution of the Navier -Stokes equation through the scalar potential  and vector potential   in the form Substituting (7) in (6), we get: From (13) we easily get ; .
rotrot grad rot rotrotgrad rot rotrot rot graddiv gard Substituting these relations in the equation of motion (8) we find This equation will be satisfied if we assume Thus, the particular solution of equation (8) can be obtained based on the particular solutions (15) and (16). From (9) and (10) it can be seen that for finding the potentials  and   we need to know the pressure p . We illustrate it on an example, when the liquid is viscous Newtonian. In this case to the system of Navier -Stokes linearized equations (8) The solution of equation (11) is of the form From (15), for finding  we get the equation   , , The solution of equation (22) corresponding to the problem under consideration, is of the form: Using (13), (17) and (19), for the components of velocity vector we get: By means of the viscosity force formula [1] we find: n n k f r p ikJ kr a n in k k J qr J qr J qr i kx n t R R n n n in f r p J kr R R a n i k J qr ikJ qr J qr i kx n t R , , x y z q q q acting on the shell as viewed from viscous liquid n n k q f r p ikJ kr a n in k k J qr J qr J qr i kx n t R R n in q f R p J kr R R a n i k J qr ikJ qr J qr i kx n t R (26) We will look for the displacements of the shell points in the form Here where  is the main determinant, ( , 1, 2,3)   sp s p are auxiliary determinants of this system. These determinants are given in [6].
We can calculate the work performed by these loads when taking the system from the deformed state to the initial undeformed state.
We look for the perturbed velocities potential  in the form: Using (30), from equation (7) we have [10]: The stationary value of the obtained function is determined by the following system of equations:    The results of calculations were given in Fig. 2    From the figures it can be seen that availability of viscous liquid leads to decrease in the value of natural frequency vibrations of the system compared to the frequency of vibrations of the system with any liquid. As can be seen from Fig. 2 with increasing the amount of longitudinal ribs, the value of the frequency parameter increases. As the inhomogeneity parameter increases in the direction of the generatrix of the shell ,  as can be seen from Fig. 3 the value of the frequency parameter increases. Furthermore, the value of the frequency parameter increases with increasing orthotropic properties of the cylindrical shell, and decreases with increasing the liquid flow rate (Fig. 4).