Nonlinear vibration of orthotropic shallow shells of the complex shape with variable thickness

Abstract

Early R-functions theory [1] combined with variational methods have been applied to linear [2] and nonlinear vibration problems [3,4] of the shallow shells theory of the constant thickness. In the present study, we first apply R-functions theory in order to investigate the geometrically nonlinear vibrations of orthotropic shallow shells of complex shape with variable thickness. Mathematical formulation is made in the framework of classical geometrically nonlinear theory of thin shallow shells. For a discretization of the original system in time, approximation of unknown functions is carried out by using a single mode approach. In order to construct a system of basic functions, the proposed algorithm includes sequence of the linear problems such as finding eigen functions of the linear vibrations of shallow shells with variable thickness and auxiliary tasks of the elasticity theory. The linear problems are solved by the R-functions method. The developed approach allows reducing the original problem to the corresponding problem of solving nonlinear ordinary differential equations (ODEs), whose coefficients are presented in analytical form. In order to solve the obtained system of ODEs the Bubnov-Galerkin method is applied. The proposed algorithm is implemented within an automated system POLE-RL [1]. Numerical examples of large-amplitude flexible vibrations of shallow orthotropic shells with complex shape and variable thickness are introduced demonstrating merits and advantages of the R-functions method. Comparison of the obtained results regarding shells with rectangular plans with the other methods confirms the reliability of the proposed method.