Nonlinear Dynamics Of Functionally Graded Plate Subjected To Aero-thermal Load

Nonlinear dynamics of functionally graded plate subjected to aero-thermal load is analyzed in this paper. Based on Reddy's third-order shear deformation plate theory and von-Karman- type large deflection nonlinear geometric relations, basic equations of functionally graded rectangular plate's elasticity are obtained. Using Hamilton's principle, elastodynamic model of simply-supported functionally graded plate subjected to aero-thermal load is established, which consists of five degrees of freedom. Transversal motion's modal function satisfying simply-supported boundary is truncated to two modes. Nonlinear ordinary differential equations of transversal oscillation of the plate are derived by using elmination methods and Galerkin's method. Material properties vary continously in the thickness direction according to simple power-law. Temperature distribution is in line with one-dimensional heat conduction law. Hypersonic aerodynamics pressure is approximated by higher-order poiston theory. Supersonic as well as hypersonic aerodynamic pressure is considered in this paper.Average equations in different internal resonance are obtained by using multi-scale method. Based on Potter's idea, basic stationary solutions are divided into six types, including trivial state, pure modes, mixed modes, traveling waves, standing waves, and modulated traveling waves. Condition for the existence and stability of each type of solutions are discussed. Paticular results of the Aluminum-Alumina functionally graded rectangular plate are given. Effects of temperature and volume fraction index on nonlinear behaviors and stabilities of the plate are analyzed. In numerical experiments only one mode is excited at the beginning, however, many kinds of response, e.g. pure-mode oscillation, mixed-mode oscillation, n-periodic oscillation, may emerge with different temperature or different volume fraction index. Numerical results are qualitatively as same as theorical results, but are quantitatively different, possible reasons are discussed in the paper.Nonlinear aero-thermal elastodynamics of the plate subjected to linear and nonlinear aero-dynamic pressure are respectively analyzed. Average equations in 1:2 internal resonance of the plate subjected to aero-thermal load are obtained by using multi-scale method. There is no stationary solution if the plate is subjected to linear aero-dynamic pressure. With nonlinear aero-dynamic pressure, if fraction index is equal to 0.2 or 5, average system has stable equilibriums and stable periodic solutions, but relevant traveling waves and modulated traveling waves in original system have not been found. Numerical results show that the response either converges to zero or diverges to infinity. Because of the special form of the truncated two-mode function of the transversal motion, discreted by Galerkin method, linear terms of the two modes are uncoupled. So before buckling, zero equilibrium is always stable, no matter which kind of aerodynamic pressure the plate subjected to. Effects of the form of truncated two-mode function on zero-equilibrium's stability are discussed detailedly in this paper.Effects of geometrical parameters and fraction index on natural frequencies of the two modes are analyzed. Coefficients of the two resonant terms have opposite sign is a necessary condition for the existence of non-zero stationary solutions in 1:2 internal resonance. Effects of geometrical parameters on coefficients of the resonant terms are discussed. Numerical results indicate that first mode's Poincare map has self-similar structure sometimes; its spectrum has a continuous, broad-band nature. In order to recognize chaos, Melnikov's method is used. Hamilton quantity is obtained and is too small to generate homoclinic orbits, so it is concluded that chaos in the sense of Smale horseshoe is absent.