Geometrically nonlinear vibrations of slender meso-periodic beams. The tolerance modeling approach

The paper deals with geometrically nonlinear vibrations of beams with periodic structure. The original 1-D model with highly oscillating coefficients based on the Rayleigh beam theory with von Karman-type nonlinearity is converted into a system of differential equations with constant coefficients. The proposed model is obtained in the framework of the tolerance modeling technique and studied numerically using Galerkin and Runge-Kutta methods. Dynamics analysis of a simply supported uniform beam carrying a system of lumped masses with both translational and rotary inertia is performed. Natural linear frequencies and modes of vibrations are determined and compared with a finite element model. Free and forced nonlinear vibrations are analyzed within the obtained model.