DISCOVERING A PATTERN IN THE FREE VIBRATIONS OF GENTLY SLOPING SHELLS OF DIFFERENT GEOMETRY IN THE CLASSIC AND REFINED STATEMENTS

The paper reports an effective numerical procedure to solve problems on the free oscillations of isotropic gently sloping shells using a spline-approximation method of unknown functions along one of the coordinate directions. By applying the proposed procedure, we have examined the resonance frequencies of the oscillations of cylindrical shells and shells of double curvature both in a square and rectangular plan. The calculations were conducted and compared based on two theories: classic (by Kirchhoff-Love) and refined (by Timoshenko-Mindlin). We have established the dependence of natural oscillation frequencies on the ratio of shell thickness and their dimensions in the plan. It has been revealed that the frequencies of free oscillations of gently sloping shells, computed in the refined statement, have lower values than the corresponding frequencies calculated in the classic statement. With the increasing thickness of the shells, the difference in the values of corresponding frequencies increases. The calculations results were compared with the frequencies computed analytically by expanding the unknown functions into a Fourier series. The comparison has allowed us to determine the optimal scope of application of each theory. It has been established that the frequencies of free vibrations of thin gently sloping shells should be computed in a classic statement. The calculation of non-thin shell frequencies (at a ratio of the thickness to the smallest size in the plan of h/a³0.05) at any geometric parameters of the shells should be performed in the refined statement. Our results have confirmed the theoretical assumptions about the importance of considering the turning angles, first, of a rectilinear element, caused by transverse offsets, in calculating the natural oscillation frequencies of the non-thin shells. The versatility and high accuracy of the spline approximation method have been confirme