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The panel structures of flight vehicles at supersonic or hypersonic speeds are subjected to combined thermal, acoustic, and aerodynamic loads. Because of the combined thermal and acoustic loads, the panel structure may exhibit nonlinear random vibration responses, such as the snap-through phenomenon and random vibrations. These unique dynamic behaviors of the panel structure under combined thermal and acoustic loads can result in serious damage or fatigue failure of the panel structures of high-speed flight vehicles. This study investigates the nonlinear random responses of thin and thick panels under combined thermal and acoustic loads. The panels are modeled based on the first-order shear deformation theory (FSDT) to account for transverse shear deformations. The von-Karman nonlinear strain–displacement relationship is used for geometric nonlinearity in the out-of-plane direction of the panel. The thermal load distribution is assumed to be constant in the thickness direction of the panel. The random acoustic load is represented as stationary White–Gaussian random pressure with zero mean and uniform magnitude over the panels. Static and dynamic equations are derived using the principle of virtual work and the nonlinear finite element method. A thermal postbuckling analysis is conducted using the Newton–Raphson method, and the dynamic nonlinear equations are solved using the Newmark-

The skin panel structures of supersonic (1.3 < M < 5.0) and hypersonic (5.0 < M < 10) flight vehicles, such as launch vehicles, guided weapons, and fighter planes, are subjected to combined aerodynamic, thermal, and acoustic loads [

Numerous studies using various numerical methods have been conducted on skin panels under combined thermal and acoustic loads. To investigate the large deflection in the random response of thermally buckled isotropic beams, the thermal load was considered as a static preload, while the acoustic load was modeled as a uniform load [

Thus far, most previous studies [

The FSDT is used to account for the transverse shear deformations of the plate, which is significant to the behaviors of thick panels. The von-Karman nonlinear strain–displacement relationship, given by (

The constitutive equation of the panel structure considering the thermal load due to uniform temperature change,

The random acoustic load in this work is modeled based on a relatively simple statistical method [

In this study, the cut-off frequency is determined using a previously reported method [

Based on the principle of virtual work, as expressed in (

The internal work using the finite element method is defined as follows:

When (

By substituting (

The present in-house code is written using the above equations and it uses the solution techniques given in the following section to obtain nonlinear static and dynamic responses of the panel.

The Newton–Raphson method is used to calculate the nonlinear static displacement of the postbuckling of the panel structure due to the thermal load. By applying the Newton–Raphson method to solve (

The nonlinear static displacement is updated as follows:

The iteration is continued until the static displacement satisfies the convergence tolerance, which is defined as follows:

The governing equation of the dynamic solution is defined as follows:

To increase computational efficiency, the Guyan reduction method [

The dynamic transverse displacement at the (

For calculating the dynamic transverse displacement, iterative calculation is performed until the displacement obtained from (

This section describes the numerical investigation, which is conducted using the nonlinear finite element method, of the nonlinear static and dynamic behaviors of the thin and thick square panels under combined thermal and acoustic loads. The elastic modulus (E), Poisson ratio (^{9} Pa, 0.3, 2763 kg/m^{3}, and 12.8 ×10^{-6°}C^{−1}, respectively. The planar dimension of the square panel is 0.3048 m, the thickness of the thin panel (a/

The calculated natural frequencies for the lowest six modes of the thin and thick panels are summarized in Table

Natural frequencies of thin and thick square panels (units: Hz).

Mode No. | Thin panel | Thick panel |
---|---|---|

1st | 53.41 | 320.02 |

2nd | 133.75 | 799.67 |

3rd | 214.04 | 1277.10 |

4th | 269.66 | 1606.54 |

5th | 349.83 | 2079.74 |

6th | 468.13 | 2774.34 |

Mode shapes of thin panel (a/h = 300).

1st mode, 53.41 Hz

2nd mode, 133.75 Hz

3rd mode, 214.04 Hz

Mode shapes of thick panel (a/h = 50).

1st mode, 320.02 Hz

2nd mode, 799.67 Hz

3rd mode, 1277.10 Hz

Before investigating the nonlinear behavior of the panels under combined loads, the generation of a random acoustic load is validated, as presented in Table _{0} and ^{−6} Pa and 1024 Hz, respectively. As can be seen in the table, the RMS values of the present random loads are quite similar to the previous results [

Validation of random acoustic load generation.

RMS | |||
---|---|---|---|

| Present | Ref. [ | Error [%] |

110 | 0.0205 | 0.0207 | −0.9662 |

120 | 0.0653 | 0.0650 | 0.4615 |

130 | 0.2064 | 0.2050 | 0.6829 |

140 | 0.6503 | 0.6496 | 0.1078 |

Time history of random acoustic load (

Figure

Thermal postbuckling response (a/h = 300).

In this section, the nonlinear dynamic behaviors of the thin panel are investigated under combined thermal and acoustic loads. Figure

Small random vibrations about flat position (a/h = 300).

Nondimensional displacement

Stress on bottom surface

Random vibrations about buckled position (a/h = 300).

Nondimensional displacement

Stress on bottom surface

Figure

Snap-through responses (a/h = 300).

Nondimensional displacement

Stress on bottom surface

This ST behavior may seriously affect the fatigue life of the panel structure. When the magnitude of the

Large random vibrations about flat position (a/h = 300).

Nondimensional displacement

Stress on bottom surface

The material and geometrical properties of the thick plate are the same as those used in the previous sections. Thus, for the thick panel, a thickness of 0.006096 m is used, with a thickness ratio (a/h) of 50. Figure

Thermal postbuckling response (a/h = 50).

The nonlinear dynamic behaviors of the thick plate are examined under combined thermal and acoustic loads in a similar manner to that of the thin panel. Unlike the previous postbuckling analysis of the thick panel, the nondimensional thermal load for the thick panel is represented using the buckling critical temperature change of the thick panel (

Small random vibrations about flat position (a/h = 50).

Nondimensional displacement

Stress on bottom surface

Random vibrations about buckled position (a/h = 50).

Nondimensional displacement

Stress on bottom surface

Snap-through responses (a/h = 50).

Nondimensional displacement

Stress on bottom surface

Large random vibrations about flat position (a/h = 50).

Nondimensional displacement

Stress on bottom surface

In this work, the nonlinear random vibration responses of thin and thick panels were investigated considering transverse shear deformations under combined thermal and acoustic loads. The panel was modeled based on the FSDT for plates, to account for the transverse shear deformations. The von-Karman nonlinear strain–displacement relationship was used for the geometric nonlinearity. The random acoustic load was assumed to be a stationary White–Gaussian random pressure with zero mean and uniform magnitude over the panel structure. The static and dynamic equations of motion were derived using the principle of virtual work and the nonlinear finite element method. The Newton–Raphson method was used for thermal postbuckling analysis. The nonlinear dynamic equations in the time domain were solved by the Newmark-

The coefficients, matrices, and vectors used in (

In-plane, in-plane-bending, and bending stiffness matrices, respectively

Panel lengths in x and y directions (m)

In-plane strain vector

Thermal load vector and random acoustic load vector

Cut-off frequency (Hz)

Thickness of a panel (m)

Mass moments of inertia (kg·m^{2})

Thermal geometric stiffness matrix

Tangential stiffness matrix

First- and second-order nonlinear static stiffness matrices

First- and second-order nonlinear transient stiffness matrices

Static-transient nonlinear stiffness matrix

Matrices of mass, proportional damping, and linear stiffness

Reduced matrices of mass, damping, and stiffness

Resultant vectors of in-plane force and moment

Effective force vector

Reference pressure (Pa)

Spectrum density and cross-spectral density function

Sound pressure level (dB)

Critical temperature change (°C)

Time step size (s)

In-plane displacement vectors in x and y directions

Transverse displacement vector in z direction

Coefficient of thermal expansion (°/C)

Rotation vectors of the normal in xz and yz planes

Density (kg/m^{3})

Strain vectors of bending and shear.

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This study was supported by the Agency for Defense Development (Assignment no. ADD-06-201-801-014).