FORCED VIBRATIONS OF A INHOMOGENEOUS ORTHOTROPIC CYLINDRICAL SHELL STIFFENED WITH A CROSS-SYSTEM OF RIBS IN LIQUID

In the paper we study forced vibrations of an orthotropic cylindrical shell inhomogeneous in thickness and stiffened with a cross-system of ribs in liquid under the action of inner radial pressure pulsating in time. Based on Hamilton – Ostrogradsky variational principle, we construct a system of equations to determine the displacements of the mid-surface points of an orthotropic cylindrical shell inhomogeneous in thickness and stiffened with a cross-system of ribs under dynamical interaction with liquid. Surface loads acting on the cylindrical shell inhomogeneous in thickness and stiffened with a cross-system of ribs as viewed from liquid are determined from the solutions of liquid motion equations written in potentials. Analytic formulas for finding the displacements of the midsurface points of a liquid-contacting orthotropic cylindrical shell inhomogeneous in thickness and stiffened with a cross-system of ribs, were obtained.

Introduction. To give more rigidity, the shells are stiffened with ribs, and this time slight increase in the mass of the construction increases its strength. Such constructions can be in contact with external medium and be subjected not only to statistical loads but also to dynamical ones. Strength analysis, stability and vibrations of such constructions play an important role when designing modern machines and apparatus. The paper [3] studies forced vibrations of an external liquid-contacting orthotropic longitudinally stiffened cylindrical shell inhomogeneous in thickness under the action of inner radial pressure pulsating in time. Using Hamilton -Ostrogradsky variational principle, the system of equations is structured to determine the displacements of the midsurface points of a liquid-contacting orthotropic cylindrical shell inhomogeneous in thickness and stiffened with a cross-system of ribs. Analytic formulas for finding the displacements of the midsurface points of an orthotropic cylindrical shell stiffened with a cross-system of ribs under the dynamical interaction with liquid, are found. In [4] a problem on forced axially symmetric vibrations of a liquid-filled isotropic cylindrical shell stiffened and loaded with axial compressive forces, was studied. The paper [3] was devoted to forced vibrations of a longitudinally stiffened homogeneous orthotropic cylindrical shell in liquid under the action of inner radial pressure. Analytic formulas are obtained for finding the displacements of the midsurface points of a liquid-contacting, longitudinally stiffened homogeneous orthotropic cylindrical shell. Forced vibrations of an orthotropic cylindrical shell with solid and liquid medium and stiffened with a cross-system of ribs under the action of external radial pressure pulsating in time, are studied in [6]. The surface loads acting on solid medium as viewed from liquid, are determined from the solutions of liquid motion equation written in potentials. Analytic formulas for finding the displacements of the midsurface points of a liquidcontacting, longitudinally stiffened orthotropic cylindrical shell are found.
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 is 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 [2] free vibrations of an orthotropic, laterally stiffened, ideal fluidfilled 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 statement. A ribbed shell is considered as a system consisting of the own anisotropic shell and longitudinal, lateral ribs rigidly connected with it along the contact line ( Fig. 1). It is accepted that the stress-strain state of the shell can be completely determined within the linear theory of elastic thin shells based on Kirchhoff -Liav hypothesis, and for calculation of ribs the theory of curvilinear bars is chosen so that the coordinate lines coincide with the principle curvature line of the shell midsurface. It is assumed that the ribs are located along the coordinate lines, and their edges are located like the edges of sheathing lie in the same coordinate plane. The strain state of the sheathing may be determined by three components of displacements of its midsurface, ,  u and w . This time, the rotation angles of normal elements 1 2 ,   with respect to coordinate lines y and x are expressed by w and  by means of the dependences 1 2 , Considering that according to the accepted hypothesis we have the constancy of radial deflections and equality of appropriate twist angles following from the conditions of rigid junction of ribs with a shell, we write the following relation: Here For external actions it is assumed that the load surfaces acting on a ribbed shell as viewed from liquid, can be reduced to normal components , z q applied to the shell midsurface.
Differential equations of motion and natural boundary conditions for a longitudinally stiffened orthotropic cylindrical shell with liquid under the action of axial compression are obtained by the Hamilton -Ostrogrdasky variational principle. For that we write in advance the potential and kinetic energies of the system.
There are various ways for accounting inhomogeneity of a shell material. One of them is that the Young modulus and density of the shell material are accepted as normal coordinate function [8]. It is assumed that the Poisson ratio is constant. In this case, the functional of total energy of elastic deformation of the orthotropic cylindrical shell is of the form: R is the radius of shells midsurface, h is shell thickness; , ,  u w are the components of displacements of the shell midsurafce points. Suppose that where 10 20 , E E are elasticity module of the shell material in coordinate directions, is the shear modulus of the shell, is the density of the homogeneous shell material.
Allowing for (3) and (4), the total energy functional of the cylindrical shell has the form: The expressions for potential energy of elastic deformation of the  i th longitudinal rib and the  j th lateral bar are as follows [1]: The potential energy of external surface loads   , , x y z q q q q and loads zz q acting as viewed from liquid and applied to the shell is determined as a work performed by these loads when taking the system from the deformed state to the initial undeformed state and is represented as: The total potential energy of the system equals the sum of potential energies of elastic deformations of the shell, longitudinal ribs and lateral ribs and also potential energies of all external loads as viewed from liquid and potential energy of radial pressure: Kinetic energies of longitudinal and lateral ribs are written as follows [1]: Here t is a time coordinate, ,   i j is density of materials from which the  i th longitudinal and the  j th lateral bar was made.
The kinetic energy of the inhomogeneous cylindrical shell stiffened with cross ribs The equations of motion of a ribbed shell were obtained based on the Hamilton -Ostrogradsky principle of stationarity of action: give them.
Since the system (19) is inhomogeneous, from it we get the displacement amplitudes: .
q a a a a q a a a a q a a a a a a a a a a a a a a a a a a a a a a Note that for 0   the displacement amplitudes go to infinity and this corresponds to the resonance case.
Numerical results. Let us consider some results of calculations carried out proceeding from the above dependences (18) of displacements using the finite element method.
The followings were accepted for geometrical and physical parameters characterizing the materials of the shell, liquid and longitudinal bars:  The dependences of 0 / w q on frequency 1  for different ratios 10 20 E E were represented in Fig. 2. Here solid lines correspond to 10 20 1, 25  E E , dotted lines to-10 20 0, 25.