Geometrically nonlinear vibration of The Vierendeel Sandwich Plate

Abstract The fourth-order geometrically nonlinear partial differential equations of motion for the Vierendeel Sandwich Plate with five generalized displacements are established considering the second-order nonlinear variables in the displacement components of the Green strain tensor. The plate is composed of an upper surface plate, inferior rib, superior rib, and short column. Considering the isotropy of the upper surface plate and the equivalent plate material of the inferior rib and superior rib, the transverse shear stress is not considered. The short column is equivalent to a transversely isotropic sandwich layer of material, and only its transverse shear stress is considered. The geometric nonlinear vibration equations of the Vierendeel Sandwich Plate are obtained by functional variation. Then, the geometric nonlinear vibration solution expressed by the elliptic function is obtained by using Galerkin's method, and the curves of the relationship between the ratio of the linear period to the nonlinear period and the amplitude under different side length ratios are obtained. The numerical simulation results of nonlinear equations show that with the superposition of modes, there are periodic, multi-periodic, and chaotic responses, indicating that these motions occur alternately. The developed theory is an extension of the plate vibration theory.