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Chaotic and periodic motions of an FGM cylindrical panel in hypersonic flow are investigated. The cylindrical panel is also subjected to in-plane external loads and a linear temperature variation in the thickness direction. The temperature dependent material properties of panel which are assumed to be changed through the thickness direction only can be determined by a simple power distribution in terms of the volume fractions. With Hamilton’s principle for an elastic body, a nonlinear dynamical model based on Reddy’s first-order shear deformation shell theory and von Karman type geometric nonlinear relationship is derived in the form of partial equations. A third-order piston theory is adopted to evaluate the hypersonic aerodynamic load. Here, Galerkin’s method is employed to discretize this continuous nonlinear dynamic system to ordinary differential governing equations involving two degrees of freedom. The chaotic and periodic response are studied by the direct numerical simulation method for influences of different Mach number and the value of in-plane load. The bifurcations, Poincare section, waveform, and phase plots are presented.

With the continuous variation of the material properties along the thickness, functionally graded materials (FGM) can be used in high temperature gradient environments especially when they are made of metal and ceramic. The metal can keep a certain extent of toughness and ceramics have superior heat resistant ability. So they usually act as thermal protection structures in spacecraft and other structural components in high temperature environments [

Since the first report of flutter instability for circular cylindrical shells, the studies of the aeroelastic stability of cylindrical shells in axial flow received extensive attention [

Librescu et al. [

When it comes to FGM cylindrical panel identification, damage detection, and the control of the dynamics, it is necessary to investigate their complex nonlinear flutter in hypersonic air flow in great detail [

In the present research, the bifurcations and chaotic dynamics of the hypersonic FGM cylindrical panel subjected to thermal and mechanical loads are investigated by applying geometrical nonlinear and the third-order piston theory. Materials properties of the constituents are graded in the thickness direction according to a power law distribution. Only transverse nonlinear oscillations of the FGM cylindrical panel are considered; the equations of motion can be reduced into a two-degree-of-freedom nonlinear system. By the numerical method, the nonlinear dynamical equations are analyzed to find the nonlinear responses of the system.

Consider a simply supported hypersonic FGM circular cylindrical panel of a length

The model of an FGM cylindrical panel with simply supported edges and the coordinate system.

Assume that the panel material is made of a composite of the ceramics and metals. The material properties of constituents of the panel including density

The effective material properties

The linear piston theory is valid for Mach numbers changing from

For a cylindrical panel, which is exposed to an external hypersonic flow field parallel to the centerline of the panel on the surface, the aerodynamic pressure

The displacement components for the FGM cylindrical panel based on Reddy’s first-order shear deformation theory [

It is assumed that the transverse normal stress is negligible and normals are not vertical to the midplane after deformation. Substituting displacement components into Von Karman nonlinear strains-displacement relations, the strains in terms of middle-surface displacements are given as

The constitutive relations of the panel in which the thermal effects due to temperature difference are considered can be written as

The thermal expansion coefficient

By using Hamilton’s principle, the motion equations in terms of midplane displacements are obtained as follows [

Thermal force resultants due to temperature rise are functions of the temperature and coefficient of thermal expansion equation. They can be calculated by

Here, the shear correction factor

For simply supported hypersonic FGM circular cylindrical panel with rectangular base, the first two mode shapes that satisfy the boundary conditions are assumed to be

Compared to the transverse inertia term, the influences of the in-plane and rotary inertia terms on the vibration of the panel are small and can be neglected; see [

All the coefficients in (

In order to validate the numerical results presented in this study, a comparison is shown in Figure

Comparison of nondimensional center deflection of the square plate with simply supported edges under a suddenly applied uniform load by Reddy’s FEM results [

In this study, Al_{2}O_{3} and Ti-6Al-4V are chosen as the two constituent materials of the FGM cylindrical panel. The properties for these two constituent materials can be found in Shen [

Firstly, the Mach number and frequency of the in-plane excitation are taken to be 5.0 and

(a) and (b) depict the bifurcation diagrams of

As the amplitude of excitation increases within region of

To better understand nonlinear dynamical behaviors, phase portraits, time responses, the Poincare section, and three-dimensional phase portrait are illustrated. Figure

The periodic response of the FGM cylindrical panel occurs when the magnitude of the in-plane excitation is taken to be

The multiple-periodic motion of the FGM cylindrical panel occurs when the magnitude of the in-plane excitation is taken to be

The chaotic motion of the FGM cylindrical panel occurs when the magnitude of the in-plane excitation is taken to be

Figure

(a) and (b) depict the bifurcation diagrams of

The chaotic motion of the FGM cylindrical panel occurs when the magnitude of the in-plane excitation is taken to be

The nonlinear dynamics of an FGM cylindrical panel under a hypersonic flow are presented. The material properties are graded continuously throughout the thickness of the panel according to the power law function and are temperature dependent. A third-order piston theory is applied for the hypersonic aerodynamic load. The Von Karman larger deflection theory in conjunction with energy approach is used to obtain the equations of motion. The bifurcation diagrams, phase portraits, time responses, and the Poincare section are employed to understand the periodic and chaotic motions of the cylindrical panel. It is obtained that, with the change of in-plane load parameter, the different nonlinear dynamics of the FGM panel occur. Different nonlinear dynamic behaviors alternate from stable periodic motion to instable chaotic motions in hypersonic flow. In addition, from the phase portraits and time responses, it can be seen that when the chaotic motion occurs, their phase portraits resemble each other in appearance for different Mach.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors acknowledge the financial support of the National Natural Science Foundation of China through Grants nos. 11272063, 11102226, and 11472298, the Science Foundation of Beijing Municipal Education Commission through Grant no. cit&tcd201304112, and Foundation of Tianjin City through Grant no. 13JCQNJC04400.