Abstract

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.

1. Introduction

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 [1, 2]. It is well known that, due to the combined load of airflow and heating, the flexible panels might exhibit large aerothermal deflections [3]. In an extensive search of panel flutter literature, a number of investigations were dedicated to FGM plates with supersonic or hypersonic flow regimes. Considered a curved skin panel with geometrical imperfection, Abbas et al. [4] gave its flutter in unsteady flow by numerical simulation and Galerkin method. With the help of structural nonlinear and the third-order piston theory, the governing equations were derived. The effects of the system parameters on the flutter were discussed in detail. Using the linear approach, the flutter of rectangular flat plates in supersonic flow and thermal environment was studied by Prakash and Ganapathi [5]. The plate was subjected to the two-dimensional aerodynamic force. They showed that under real flight conditions heating caused by aerodynamics is enormous. Sohn and Kim [6] took a static and dynamic stability study on the panel under aerodynamic force as well as thermal loads. Ibrahim et al. [7, 8] investigated the thermal buckling and nonlinear flutter of thin FGM panels under the action of aerothermoelasticity by the finite element method. Then they presented the results for different factors. Accounting for both the geometric and aerodynamic nonlinearities, Prakash et al. [9] studied the nonlinear flutter of FGM plates under high supersonic airflow in frequency domain and time domain, respectively. The influence of various parameters including the geometrics and physics on the flutter of FGM plates was discussed. Hosseini and Fazelzadeh [10] used the numerical and analytical methodologies to study postcritical and vibration behaviors of the FGM panels in a supersonic air flow. Ibrahim et al. [8] analyzed nonlinear flutter and thermal buckling of an FGM panel under the combined effect of elevated temperature conditions and aerodynamic loading. Lee and Kim [11] dealt with the aerothermopostbuckling behaviors of the FGM panel in supersonic air flow. Limit-cycle vibration was found in this study by Newmark time integration method and Guyan reduction technique. Mey et al. [12] found that, in many studies which are about the flutter characteristics of FGM plates, the linear or nonlinear theory in conjunction with the first-order piston theory used to approximate the aerodynamic pressure was considered. The first-order piston theory is valid for sufficiently high supersonic Mach numbers on surfaces with small geometric characteristics.

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 [13]. Numerous studies on the cylindrical shells focused on their flutter. Marzocca et al. [14] reviewed the corresponding researches that analyse the dynamic behavior of curved and flat panels exposed to supersonic flow fields. Hosseini et al. [15] analyzed the nonlinear response of FGM curved panels in high temperature supersonic air flows. The effects of curved panel height-rise and volume fraction index on the nonlinear dynamical behavior of the panel which is subjected to aerothermoelastic loads are investigated.

Librescu et al. [16] presented a theoretical investigation of the flutter and postflutter of the long thin-walled circular cylindrical panels in supersonic/hypersonic flow field. For the problems of the flutter boundaries about the simply supported functionally graded truncated conical shell subjected to supersonic air flow, Mahmoudkhani et al. [17] made aerothermoelastic analysis to predict the flutter boundaries. The flutter boundaries were obtained for the FGM conical shells with different semivertex cone angles, different temperature distributions, and different volume fraction indices.

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 [18]. However, to the best of the authors’ knowledge, works on dynamic instability of FGM cylindrical panel subjected to supersonic/hypersonic flow, including the effects of thermal load and in-plane loads, appear to be scarce in the open literature. Many interesting researches used the linear shell theory. There are a few literatures on the nonlinear dynamic behavior of FGM cylindrical panel taking into account the aerodynamic nonlinearities.

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.

2. Theoretical Formulation

2.1. Model of the FGM Cylindrical Panel

Consider a simply supported hypersonic FGM circular cylindrical panel of a length , thickness , midsurface radius , and angular width . This panel is also subjected to the in-plane harmonic excitations. Cartesian coordinate is adopted to describe the deformations of the FGM circular cylindrical panel. The coordinates and are in longitudinal and tangential directions, respectively. The coordinate is taken to be positive outward radially, as shown in Figure 1. The displacements of an arbitrary point are denoted by , and . Assume that , and are the displacements of a point in the middle plane in the axial, circumferential, and radial directions, respectively. The rotations of the transverse normal to the midplane about and axes are assumed to be and , respectively. The in-plane excitation of the FGM panel distributed uniformly along the direction at the end of panels and is of the form , where is the static in-plane preload and is part of time dependent that has the form . Here, is the frequency of excitation along axial direction.

Figure 1 

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 , elastic moduli , and thermal coefficient of expansion are temperature dependent and can be expressed as [19] where , and are temperature dependent coefficients and is the environment temperature.

The effective material properties vary continuously in the direction by following a simple power law in terms of the volume fractions and can be expressed as where and indicate, respectively, the properties of the ceramic and metal and and are their volume fractions and have the following relationship: the metal volume fraction can be written as where is the volume fraction exponent which can depict the material variation profile through the thickness. According to Zhang et al. [20], linear temperature change is considered which has the form where and indicate the temperature of the top and bottom surfaces of the panel, respectively. The FGM circular cylindrical panel is subjected to a uniform temperature variation , where is the reference temperature.

2.2. Aerodynamics Loading

The linear piston theory is valid for Mach numbers changing from to 5 and for higher Mach numbers the nonlinear piston theory must be used in Mei et al. [21]. To study the nonlinear oscillation of functionally graded material cylindrical panel under a hypersonic air flow, one should use third-order piston theory. Piston theory aerodynamics, which is used in problems of oscillating airfoils advanced by Lighthill [22] and later used as an aeroelastic tool by Ashley and Zartarian [23], is a popular modeling technique for supersonic and hypersonic aeroelastic analyses.

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 using the third-order piston theory is expressed by Amabili [13] and Cao and Zhao [24] as where is the adiabatic exponent and and are the Mach number and time, respectively. The free-stream static pressure is given as where , and are the free-stream air density, velocity, and the free-stream speed of sound, respectively.

2.3. Geometry and Constitutive Relations

The displacement components for the FGM cylindrical panel based on Reddy’s first-order shear deformation theory [25] can be represented as

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 where are the elastic constants which can be expressed as

The thermal expansion coefficient can be given as

2.4. Equations of Motion

By using Hamilton’s principle, the motion equations in terms of midplane displacements are obtained as follows [20]: where is the damping coefficient, all kinds of the stiffness elements , and of the FGM cylindrical panel are denoted by and various mass inertia terms in (12) can be defined as

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 that is introduced by Reddy [25] and Kadoli and Ganesan [26] is equal to .

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 where and are the time dependence amplitudes of the first two modes. The constant can be represented by .

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 [27, 28]. Following the work given in [20], the displacement components of can be expressed in terms of . Applying the Galerkin method on (12), one can obtain a coupled set of nonlinear ordinary differential equations in time that can take the form as follows: where and . The coefficients and are the results of thermal stress resultants and the static components of in-plane preloads. The variables and are related to the aerodynamic pressure; they can be expressed as