VIBRATIONS REINFORCED BY LONGITUDINAL RIBS OF AN INHOMOGENEOUS ORTHOTROPIC CYLINDRICAL PANEL CONTACTING WITH A VISCOUS-ELASTIC MEDIUM

Vibrations of an orthotropic cylindrical panel, non-uniform in thickness, supported by longitudinal ribs, lying on a linearly viscoelastic foundation, were investigated. The Hamilton – Ostrogradsky variational principle was used to find the vibration frequencies of the panel. A frequency equation is constructed, its roots are found, and the influence of physical and geometric parameters characterizing the system is studied.

Cylindrical panels are widely used in modern technology, power engineering, and in various fields of construction and engineering. In many cases, depending on production technology and a number of various reasons, mechanical properties of the material of cylindrical panels become continuously inhomogeneous along the length of the panel. In operational conditions these panels are in contact with different nature medium and they are stiffened when it is necessary.
In [5], a problem of lateral vibrations of a circular cross-section inhomogeneous cylindrical shell lying on a viscous-elastic foundation, is considered. It is assumed that the modulus of elasticity and density are continuous functions of thickness coordinate. The problem of natural vibrations of a circular cross-section cylindrical shell inhomogeneous only along the length and lying on inhomogeneous viscous-elastic medium, is considered in [6,7]. The solution of the problem is reduced to the system of two linear differential equations with respect to the stress function and deflection. The method of separation of variables and the Bubnov -Galerkin method is used when solving the problem.
The paper [7] is devoted to vibrations of an orthotropic, cylindrical panel inhomogeneous in thickness, stiffened with lateral ribs and lying on a linearly viscous-elastic foundation. Using the Hamilton -Ostrogradsky variational principle for finding vibrational frequencies of a cylindrical panel inhomogeneous in thickness, stiffened with lateral ribs and lying on a linear elastic foundation, the frequency equation was constructed, its roots were found and the influences of physical and geometrical parameters characterizing the system, were studied. The paper [2] was devoted to free vibrations of a flowing fluid-contacting, isotropic, inhomogeneous cylindrical shell stiffened with cross system of ribs. Using the Hamilton -Ostrogradsky variational principle, the system of equations of motion for a flowing fluid contacting anisotropic cylindrical shell inhomogeneous in thickness and stiffened with cross systems of ribs was solved. The paper [4] deals with natural vibrations of a soilcontacting cylindrical shell stiffened with annular ribs and subjected to compressive forces.
To apply the Hamilton -Ostrogradsky variational principle, we write the total energy of the structure under investigation, consisting of an orthotropic cylindrical panel inhomogeneous in thickness and stiffening elements whose number varies. Furthermore, from the inside, the structure is in contact with a viscous-elastic medium (Fig. 1). To take into account inhomogeneity of a cylindrical shell in thickness, we will proceed from three-dimensional functional. In this case, the functional of total energy of the cylindrical shell is of the form     There are various ways for taking into account inhomogeneity of the shell material. One of them is that the Young modulus and density of the material are accepted as functions of the normal coordinate z [3]. It is supposed that the Poisson ratio is constant. In this case, the strain-stress ratio is of the form            .
In ( The kinetic energy of ribs are written in the form [8] In expressions (5) Potential energy of external surface loads acting as viewed from elastic medium, applied to the shell is determined as a work performed by these loads when taking the system from the deformed state to the initial not deformed one and is represented as follows Suppose that the plate lies on viscoelastic base, where reaction z q is connected with flexure w in the following relation: The total energy of the system equals the sum of energies of elastic deformations of the shell and lateral ribs, and also potential energies of all external loads acting as viewed from viscous-elastic medium Let the plate be comprehensively fixed with hinges. Then the following boundary conditions should be fulfilled: The frequency equation of a ridge, inhomogeneous, orthotropic, flowingfluid contacting shell was obtained on the base of Ostrogradsky -Hamilton principle of stationarity of action Jdt is Hamilton action,  t and  t are the given arbitrary moments of time.
Complementing the total energy of the system (9) with contact (10), we get a problem of natural vibrations of a viscous-elastic medium-contacting orthotropic cylindrical shell inhomogeneous in thickness and stiffened with lateral system of ribs. In other words, the problem of natural vibrations of a viscous-elastic medium-contacting orthotropic, cylindrical shell inhomogeneous in thickness and stiffened with cross-system of ribs is reduced to integration of expressions for the total energy of the system (9) In expression (9) , ,  r u w are variable values. These unknown values are approximated in the following way: Using (12) for w , we can calculate the work 0 A when transferring the system from the deformed state to the initial undeformed state The potential and kinetic energies of elastic deformation of the i -th longitudinal ribs are follows:  sin .
After integrating the expression for the potential and kinetic energy of elastic deformation of the  i th longitudinal ribs in time t from  t to  t , we obtain:   Using (12) for w , we can calculate the total energy V of the cylindrical shell After integrating the expression for the total energy V of the cylindrical shell: in time t from  t to  t , we obtain:   Using the expressions (13), (14) and (15) for the complete energy of the system under study, we obtain: Substituting (16) in (9), after integration we get a function of variables 0 0 0 , ,  u w . The stationary value of the obtained function is determined by the following system The non-trivial solution of the system of third order linear algebraic equations is possible only in the case when  is the root of its determinant. The definition of  reduces to the algebraic equation Here the elements ij a look like:  The result of calculations were given in Fig. 2 in the form of dependence of the frequency  on the amount of stiffening bars 2 k on the shell surface, in Fig. 3 in the form of dependence of frequency  on in homogeneity parameter  . The figure shows that with increasing the amount of lateral ribs, the value of the vibrations frequency of the system increases. Fig. 3 shows that with increasing the inhomogeneity parameters the vibrations frequencies of the system also increase. УДК 539.3