Geometric nonlinear vibration theory of the Vierendeel Sandwich Plate based on generalized variational method

• Published in: Archive of applied mechanics (1991) Vol. 94; no. 6; pp. 1667 - 1689
• Main Authors: Liu, Dingyuan, Kuang, Kaicong, Lu, Yaqin, Ma, Kejian
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2024; Springer Nature B.V
• Subjects: Classical Mechanics; Elliptic functions; Engineering; Equations of motion; Equivalence; Free vibration; Galerkin method; Independent variables; Interlayers; Isotropic material; Mathematical analysis; Nonlinear differential equations; Nonlinear equations; Original; Orthogonality; Partial differential equations; Strain; Surface layers; Tensors; Theoretical and Applied Mechanics; Thin plates; Nonlinear; Vibration; Generalized variational method; Vierendeel Sandwich Plate; Chaotic response
• ISSN: 0939-1533; 1432-0681
• DOI: 10.1007/s00419-024-02605-6

The fourth-order geometrically nonlinear partial differential equations of motion for the Vierendeel Sandwich Plate (VSP), with five generalized displacements, are established considering the second-order nonlinear variables in the displacement components of the Green strain tensor. Considering the material orthogonality of the equivalent thin plate with the upper surface layer, upper and lower ribs of VSP, the short column is equivalent to the transverse isotropic interlayer of the material, and the equivalent material parameters of the upper and lower ribs of the plate are derived using mechanical methods. An energy function is established by treating strain as an independent variable, and the geometric nonlinear vibration equation of VSP is obtained through functional variation. Then, the free vibration solution expressed by the elliptic function is obtained using the Galerkin method, and the accuracy of the numerical results of the nonlinear equation is verified by examples. The numerical simulation obtained curves on the relationship between the ratio of the linear period to the nonlinear period and amplitude under different half-axis ratios. It was found that the Orthogonal Anisotropic VSP exhibits alternating periodic, multi-periodic, and chaotic response motions with the superposition of modes. This theory can help to an in-depth understanding of VSP's vibration.