Abstract
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 confirmed.
Highlights
The isotropic gently sloping shells, rectangular in plan, are widely used in many spheres of human activity, from aviation to construction
An effective numerical procedure has been proposed for solving the problems on the free oscillations of gently sloping isotropic shells in order to perform calculations involving the application of a spline-approximation method of unknown functions
We have devised a numerical procedure for solving the problems on the free oscillations of gently sloping isotropic shells, rectangular in plan, with the application of a spline-approximation method of unknown functions and a numerical method of discrete orthogonalization in a combination with the incremental search method
Summary
The isotropic gently sloping shells, rectangular in plan, are widely used in many spheres of human activity, from aviation to construction. When calculating the thin gently sloping shells, it is advisable to construct a computation algorithm based on the classic theory by Kirchhoff-Love. It implies the introduction of a series of simplifications to the initial equations of the elasticity theory, which, in this case, exert an insignificant influence on the calculation results but considerably simplify the resulting equations. When calculating the non-thin shells, it is expedient to take into consideration the turning angles of an initial rectilinear element caused by the transverse offsets This clarification is taken into consideration in the theory of the Timoshenko-Mindlin type. 2/7 ( 104 ) 2020 more expedient, in terms of accuracy of the obtained results, to compute the natural oscillation frequencies of gently sloping shells in the refined statement. Determining such a limit of the shell thickness would make it possible to properly choose between the classic and refined statements when investigating the natural oscillation frequencies of gently sloping shells, which could improve accuracy
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