Investigation of free vibrations of three- layered cylindrical shell supported by transverse ribs

Mykola Surianinov , Tetiana Yemelianova , Dina Lazarieva *3 Department of applied mathematics, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine sng@ogasa.org.ua Department of physic and general engineering disciplines, Kherson State Agriculture University, Kherson, Ukraine e.tatyana.2014@ukr.net *Department of structural mechanics, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine dvl7@ukr.net


I. INTRODUCTION
Layered constructions, in particular three-layered shells, are widely used in many industries, including industrial and civil construction.
Three-layered shell consists of two comparatively thin external layers of strong material between which comparatively thick layer of material with low strength and light volume weight is placed. External layers are called bearing and internal layer is called aggregate.
Three-layered constructions are classified by the type of aggregate, bearing layers' material, connection type. Light-weight aggregates can be styrofoam, honeycomb core, corrugations (single or double), tubular aggregate, etc.
Three-layered construction application allows to increase the shell stability, reduce the negative influence of initial deflections on stability, use mechanical properties of material in better way.

II. PROBLEM FORMULATION
Forsuch kind of constructions, the solution of free vibrations problem has fundamental importance because it allows to solve many other problems of dynamics. A special role has information about the first frequency of free vibrations. The problem of three-layered shell without support by ribs was considered by a lot of authors. Let's note the works that have already become classical [1][2][3][4], as well as later publications [5][6][7][8][9]. There are no too much publications devoted to vibrations of three-layered shells supported by ribs [10][11][12].

A. Aim of paper
The aim of the work is to build a computational model and to develop an algorithm for studying the free vibrations of a three-layer cylindrical shell with a lightweight aggregate that is supported by transverse ribs with edges simple support.

B. Materials and methods
Using the functional-action by Ostrogradskiy-Hamilton, there is received variation equation of transverse vibrations of three-layered shell of symmetric construction, that is supported by ribs in two perpendicular directions, taking into account the action of longitudinal forces in middle planes of external layers and in ribs.
For external bearing layers of shell there are accepted hypotheses of Kirchhoff-Love and for aggregate there is accepted the linear law of tangential displacement on thickness change. Transverse deformations of aggregate was not taken into account. For plates ribs the hypotheses of Bernoulli was accepted. Only ribs bending in vertical plane was taken into account. With the help of the limit transition, conditions are obtained for the ribs lines without considering the shear deformations in the ribs [3].

III. RESEARCH RESULTS
Let's consider free vibrations of three-layered sloping cylindrical shell with light-weight transverse-isotropic aggregate, that is supported by transverse ribs. Distance between ribs and ribs stiffness we will consider as equal (Fig. 1). Differential equations of flexural vibrations of shell area that is located between ribs   10 , has a form: In work   11 by the way of introduction of displacement function   , F x y it is performed the simplification of differential equation     1 -3 system and there is get the resolving equation of free vibrations: In equations   1 -(4) there are accepted symbols: (3) and (4) for area of shell between two adjacent ribs we will find in the form:

Solution of equations
Substituting (5) into (3) and (4), we obtain Here Assuming that diaphragms are installed on the edges of the shell, the boundary conditions for the case of free support will be written in the form: Assuming for every area its coordinate axes   12 , we will locate them in the start of area and denote   Using these conditions, we express through them values of arbitrary constant C i of solutions (6) and (7), which are determined from the system of equations (11): In equations (11) there are denoted: Conditions on line of k edge taking into account different directions of axesхfor contiguous areas that are received from variation equation we will write in form: ;       , which are included into this system must satisfy conditions of solution periodicity: Equating determinant, which consists of coefficients at A, B, C, M, E, to zero we will get frequency equation of three-layered sloping cylindrical shell, which is supported by regular transverse ribs, at edge simple support.
Parameter of the first frequency we will find by solving the transcendent frequency equation. In the tables 1-4 there are given values of parameter of first frequency of free vibrations m  of sloping cylindrical shell, that is supported by one and three ribs. Value m  was determined with taking into account the Reisner's edge effect (top row of tables) and without it (lower row of tables).If we put in the frequency equation 0 0, k  we will get the frequency equation for one-layered supported shell (with bending stiffness * 2 2 D BH  ).