A Note on Analysis of the Natural Vibrations of Shallow Shells with Cuts on the Surface

A new approach is proposed to study the dynamic behavior of shells with an arbitrary planform weakened by surface cuts (cracks.) The approach is based on the method of R-functions. The computer simulation is carried out using the POLE-SHELL problem-oriented system, which implements the method of R-functions and variational methods. Numerical results are presented     

Nonlinear free vibrations of multilayer shallow shells with a symmetric structure and with a complicated form of the plan

We propose a method to study free nonlinear vibrations of multilayer shallow shells with a complicated form of the plan. The mathematical statement of the problem is realized in the frame of a refined firstorder theory of the Tymoshenko type. A distinctive specific feature of the work is the application of the theory of R -functions and variational methods to the determination of eigenfunctions used as the basis in the construction of the required solution of a nonlinear problem. The proposed method is tested, and some new problems are solved. As a result, we constructed the amplitude–frequency dependences for spherical shells with a complicated form of the plan.     

R-functions in problems of bending of three-layer plates

We study the application of the method of R-functions to the solution of problems of bending of elastic three-layer plates of arbitrary shape. We obtain new solution structures for plates that are freely supported over the whole boundary. We solve test problems and compare the experimental and theoretical results. All numerical results were obtained using the POLE programming system.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 77–79.     

Nonlinear vibration analysis of laminated shallow shells with clamped cutouts by the R-functions method

In present work, an effective method to research geometrically nonlinear free vibrations of elements of thin-walled constructions that can be modeled as laminated shallow shells with complex planform is applied. The proposed method is numerical–analytical. It is based on joint use of the R-functions theory, variational methods, Bubnov–Galerkin procedure and Runge–Kutta method. The mathematical formulation of the problem is performed in a framework of the refined first-order shallow shells theory. To implement the developed method, appropriate software was developed. New problems of linear and nonlinear vibrations of laminated shallow shells with clamped cutouts are solved. To confirm reliability of the obtained results, their comparison with the ones known in the literature is provided. Effect of boundary conditions is studied.     

Studying the free vibrations of multilayer plates with a complex planform

An efficient method is developed to solve the free-vibration problems for arbitrarily shaped orthotropic multilayer plates in a refined formulation. The method is based on the R-function and Ritz methods. Sequences of coordinate functions satisfying kinematic boundary conditions are constructed in an analytic form. The method is used to solve the vibration problem for multilayer square and arbitrarily shaped plates. The results obtained for square plates are analyzed. A comparison of these results with those available in the literature demonstrates the efficiency of the method __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 119–125, January 2006.     

Free vibrations of flexible shallow shells of complex shape

A method is proposed for studying the free vibrations of flexible shallow shells with a complex planform. The method is based on variational and R-function methods. The R-function method allows constructing a system of basis functions in an analytic form. This makes it possible to reduce the Donnell-Mushtari-Vlasov equations to Duffing equations. The amplitude-frequency characteristics of shallow shells with a complex planform are given for different curvatures and boundary conditions. The results obtained are compared with published results for simply supported square shells to demonstrate the reliability and efficiency of the method __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 99–109, April 2007.