The bifurcation analysis on the circular functionally graded plate with combination resonances

Abstract

The bifurcation and chaos of a clamped circular functionally graded plate is investigated. Considered the geometrically nonlinear relations and the temperature-dependent properties of the materials, the nonlinear partial differential equations of FGM plate subjected to transverse harmonic excitation and thermal load are derived. The Duffing nonlinear forced vibration equation is deduced by using Galerkin method and a multiscale method is used to obtain the bifurcation equation. According to singularity theory, the universal unfolding problem of the bifurcation equation is studied and the bifurcation diagrams are plotted under some conditions for unfolding parameters. Numerical simulation of the dynamic bifurcations of the FGM plate is carried out. The influence of the period doubling bifurcation and chaotic motion with the change of an external excitation are discussed.

Thermo‐mechanical nonlinear stability analysis of geometrically imperfect functionally graded plates with microstructural defects using logarithmic structure kinematics: An unified expression

“…An exact solution for the buckling behavior of FGP is presented by Thai and Choi [14] using two variable refined plate theory. Hu and Zhang [15] employed geometric nonlinearity and the temperature-dependent properties of the materials to investigate the bifurcation and chaos of clamped circular FGPs. Shariat and Eslami [16,17] introduced the rectangular FGP with geometric defects for the perfect and imperfect plate under mechanical and thermal loading using polynomial higherorder shear theory (TSDT).…”

Combination resonances of spinning composite shafts considering geometric nonlinearity

“…Dimentberg and Bucher [14] analyzed the instability characteristics of a two-degree-of-freedom system subjected to a periodic excitation using the Krylov-Bogoliubov averaging method. Hu and Zhang [15] analyzed the bifurcation and chaos in a circular plate having clamped supports. The plate was assumed to be made of functionally graded materials and subjected to a lateral periodic force and thermal loads.…”

Magneto-elastic dynamics and bifurcation of rotating annular plate *

“…For a functionally graded circular plate, Hu et al [16] investigated unfolding problems of bifurcation equation, and plotted bifurcation diagrams. Hu and Zhang [17] analyzed bifurcation of the circular functionally graded plate with combination resonances. In response to geometrically nonlinear problem of a circular plate, Touzé et al [18] derived ordinary differential equations by using Galerkin method, and plotted the bifurcation diagrams and Poincaré maps.…”