Natural vibrations of heated functionally graded cylindrical shells with fluid

The paper presents the results of studying the natural vibrations of heated cylindrical circular FGM shells containing a quiescent ideal liquid. The temperature dependent effective properties of the material representing a mixture of zirconium oxide and titanium alloy vary through the thickness of the shell according to the power law. The distribution of temperature along the radial coordinate is determined by solving a quasi-linear one-dimensional heat conduction equation. A mathematical formulation of the problem is based on the classical theory of shells and the principle of virtual displacements. The behavior of the liquid is described in the framework of the potential theory. The corresponding wave equation together with the impermeability condition and boundary conditions are transformed into a system of equations using the Bubnov-Galerkin method. As a result, the solution of the problem, which is sought with the use of semi-analytical version of the finite element method, reduces to calculations of complex eigenvalues of the coupled system of equations. The reliability of the results, obtained by application of the developed algorithm, is verified through a comparison with the known numerical-analytical solutions. The data obtained for circular cylindrical shells with different boundary conditions have revealed the dependence of the minimal vibration frequency on temperature at different volume fractions of FGM. The critical values of temperature have been determined for different heating regimes and geometrical dimensions. The difference between the dynamic properties of empty and liquid-containing shells caused by heating has been analyzed. It has been shown that in the case of cantilevered shells the presence of liquid inside the shell exerts the most notable effect on the vibrational behavior of the system.