Nonlinear Random Response Analyses of Panels Considering Transverse Shear Deformations under Combined Thermal and Acoustic Loads

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 highspeed 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-ofplane 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-β time integration method. In the present numerical analyses, the snap-through responses for both the thin and thick panels are investigated, and the results indicate that the loading conditions that cause snap-through are different for thin and thick panels.


Introduction
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 [1].The thermal loads due to aerodynamic heating cause thermal buckling and postbuckling of the panel, which result in a sudden change in vehicle configuration.Additionally, acoustic loads due to engine noise in high-speed flights may induce random vibrations of the panel structures.When these loads are applied simultaneously, the dynamic behavior of the skin panels becomes very complex and may result in serious damage, such as fatigue failure, of the panels of high-speed flight vehicles.Particularly, under the action of combined thermal and acoustic loads, the snap-through response of the panel structures exerts a highly important effect on the fatigue of the skin panels of high-speed vehicles.
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 [2].In the study, thermal postbuckling problems were solved to determine the deflections and stresses due to the thermal load.These deflections and stresses were then used as the initial deflections and stresses for random vibration analysis.Finite element formulation was used as an analysis method.The random vibration and fatigue life of composite panels under combined high thermal, aerodynamic, and acoustic loads were investigated [3].In that work, a shape memory alloy (SMA) hybrid composite panel, as well as isotropic and composite panels, was considered.In addition, the panel was modeled as a thin plate, and the effects of rotary inertia and transverse shear deformation were not considered.The random responses of the panel were classified into four types: large random vibration about a flat position (LV, w/h > 2.0), small random vibration at a flat position (SV, w/h < 0.2), snap-through (ST), and random vibration at a buckled position (VBP).Nonlinear vibration responses and fatigue life estimation were investigated for arbitrary composite laminated panels [4].The acoustic load was modeled using the equivalent band-limited white-noise sound pressure level (EWSPL) excitation method.The governing equations for the panel structure were represented by the finite element method, and the random responses in the time domain were calculated using the Runge-Kutta method and the Monte Carlo numerical analysis of EWSPL.In addition, the strain and stress of the panel were calculated using the switching Markov processes approach.The nonlinear responses of thin isotropic and composite plates were investigated under increased heat load [5].The plate was modeled using the finite element method for a quarter plate.The displacement and strain of the plate were investigated with regard to three behaviors (linear vibration, snap-through motions, and nonlinear random response) by using a modal reduction method to reduce the modes used for the time domain analysis.The random vibration response of composite panels under a random acoustic load and a uniform thermal load was investigated [6].The panels were represented using the first-order shear deformation plate theory.However, transverse displacement was calculated by excluding the effects of rotary inertia and shear deformations of the panels.Nonlinear random vibration responses were analyzed using the autoregressive moving averaging (ARMA) method.Nonlinear random vibration and snap-through behaviors under combined thermal and acoustic loads were investigated for a functionally graded material (FGM) panel structure consisting of a mixture of nickel and silicon nitride [7].A nonlinear finite element method was used to model the FGM panel structure in which heat and random acoustic loads were applied.The first-order shear deformation theory (FSDT) for plates and von-Karman's relationship were applied to the structural modeling, and the nonlinear random vibration response was investigated under various loading conditions.Thus far, most previous studies [2-5, 8, 9] on random vibration analysis under combined thermal and acoustic loads have used thin panel structures modeled with the classical plate theory, which does not consider rotary inertia and transverse shear deformations of the panel structure.However, the panel structures of high-speed vehicles may have heat shielding layer and stiffeners; thus, their effective thickness may not be thin.Therefore, the thin plate model used in the previous works [2-5, 8, 9] may not be appropriate for modeling of panel structures of high-speed vehicles, and the modeling of thick panels, as well as that of thin panels, is strongly required to investigate nonlinear responses under combined thermal and acoustic loads.This study considers both thin and thick panels under combined thermal and acoustic loads to overcome the drawback of the previous studies.The FSDT for plates is applied to account for the rotary inertia and transverse shear deformations of the panel structure, which are important for the behaviors of the thick plates.The present plate model is relatively simple as compared to the real panel structures of high-speed vehicles.However this simple structural model is enough for the preliminary study to investigate nonlinear random responses under combined thermal and acoustic loads in case that the effective thickness of the panel is thick.The von-Karman nonlinear strain-displacement relationship is used to express geometric nonlinearity.The equations of motion are derived by the principle of virtual work and nonlinear finite element method.The equations of motion are divided into nonlinear static and dynamic equations.The Newton-Raphson method, which is a thermal postbuckling analysis, is used for nonlinear static analysis, and the Newmark- method is used to examine nonlinear dynamic behaviors in the time domain.Using the in-house code with these numerical methods, the nonlinear random responses of both thin and thick panel structures are investigated under various combined thermal and acoustic loading conditions.This study presents four nonlinear response results, namely, LV, SV, ST, and VBP.Among these, as the ST response has a considerable influence on the fatigue life of the panel, its loading condition and stress response are investigated thoroughly.In addition, the predictions of nonlinear random responses and stresses for thin and thick panels are compared under various loading conditions.

Constitutive Equations of Panel under Combined Loads.
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 (1a) and (1b) is used to account for the geometric nonlinearity of the panel for outof-plane deformation.
The constitutive equation of the panel structure considering the thermal load due to uniform temperature change, Δ, in the thickness direction is expressed as follows: where the thermal loads are defined by (3) as follows: The random acoustic load in this work is modeled based on a relatively simple statistical method [3,4,8,9].It is assumed that the random acoustic load acts as a uniformly distributed load on the panel structure and that the mutual spectral density function has a Gaussian normal distribution.The mutual spectral density function is defined using (4), as follows: In this study, the cut-off frequency is determined using a previously reported method [4].Finally, the random acoustic load, , is formulated using a random number generation function, i.e., randn, in MATLAB [3], as follows: where  is the number of time steps for time integration.

Derivation of the Equations of Motion.
Based on the principle of virtual work, as expressed in ( 6), the equations of motion for the panel under combined loads are derived as follows: The internal work using the finite element method is defined as follows: where the displacement vector is defined as The external work is expressed with the finite element method as follows: When ( 7) and ( 8) are substituted into (6), the governing equations for the panel under combined thermal and acoustic loads can be derived as follows: where the structural damping matrix C is defined by proportional damping and added directly to the governing equation.The solution of the governing equation, expressed by ( 9), can be assumed as the sum of the nonlinear static and dynamic solutions as follows: By substituting (10) into ( 9), the static and dynamic equations of motion are obtained as follows: 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.

Solution of Nonlinear Static Equations. 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 (11a), the incremental form at the -th iteration is expressed as follows: 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: where Δ  is the difference of the static displacement vector.

Solution of Nonlinear Dynamic
Equations.The governing equation of the dynamic solution is defined as follows: To increase computational efficiency, the Guyan reduction method [10,11] is used with (15) as follows: where the upper bar represents the reduced matrices and the vector by the Guyan reduction method.The Newmark- time integration method [12] is used for calculating dynamic displacement in (16).For the time integration, the initial velocity vector ẇ 0 is zero, and the initial displacement vector w  0 is assumed appropriately.The initial acceleration vector ẅ 0 is defined as follows: The dynamic transverse displacement at the (i+1)th time step can be calculated as follows: where where the detailed definitions of coefficients, matrices, and vectors in ( 18) and ( 19) are given in the Appendix.
For calculating the dynamic transverse displacement, iterative calculation is performed until the displacement obtained from (18) satisfies the convergence condition in (20).

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 Δw
where Δw +1  is the difference in dynamic transverse displacement vector and is defined as follows:

Numerical Results and Discussion
3.1.Natural Frequency.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 (v), density (), and thermal expansion coefficient () of aluminum are 70 ×10 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/h thin = 300) is 0.001016 m, and the thickness of the thick panel (a/h thick = 50) is 0.006096 m.In the finite element model, a uniform 5 × 5 finite element mesh with nine-node elements is used.Reduced integration is applied to the integration of a transverse shear stiffness matrix to prevent the transverse shear-locking problem.
The calculated natural frequencies for the lowest six modes of the thin and thick panels are summarized in Table 1.Figures 1 and 2 show the mode shapes for the lowest three modes of the thin and thick panels.In addition, the critical temperature changes of the thin and thick panels, Δ ,thin and Δ ,thick , are calculated as 1.098 ∘ C and 39.466 ∘ C, respectively  4) are 20×10 −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 [7].In addition, Figure 3 shows an example of the generated random acoustic loads for SPL = 110 dB in the time domain.

Thermal Postbuckling Analysis.
Figure 4 shows the thermal postbuckling analysis for the thin square panel.The material and geometrical properties are described in Section 3.1.For this thin panel, the thickness ratio (a/h) is 300.As shown in Figure 4, the nondimensional maximum deflection (W max /h) of the panel increases monotonically after thermal buckling occurs and as the temperature increases.

Random Vibration Analysis.
In this section, the nonlinear dynamic behaviors of the thin panel are investigated under combined thermal and acoustic loads.Figure 5 shows the time responses of the transverse displacement and stress under a random acoustic load at an SPL of 80 dB and without thermal load (ΔT = 0).In Figure 5(a), the SV (small random vibration, w/h < 0.2) about the flat position is observed under this loading condition.Note that the displacement shown in Figure 5(a) is the total displacement defined in (10).Figure 5(b) shows the stress ( xx ) response at the center of the bottom surface of the panel.As can be seen in the figure, the magnitude of the stress response is small in comparison to that of the thin plate, which will be given later.However, when the temperature increases (Δ/Δ ,thin = 2.75), while the magnitude of the acoustic load is the same as that at an SPL of 80 dB, the response of VBP of the panel is as shown   in Figure 6(a).The displacement is damped out and goes to the buckled position due to structural damping.In addition, as the panel is buckled in the downward direction, the tensile stress response at the center of the bottom surface is shown in Figure 6(b).Figure 7(a) shows the ST response of the thin panel under the thermal load (Δ/Δ ,thin = 2.75) and increased acoustic load (SPL = 110 dB).The panel shows random vibrations in the ST response alternately at the positive and negative deflected positions in the transverse direction.Figure 7(b) shows the stress response at the center of the bottom surface for the ST response.The stress response alternates between the two mean values for the two thermally buckled states.On the bottom surface, the tensile stress indicates that the panel is buckled in the downward direction; however, the compressive stress shows that the panel is buckled in the upward direction.
This ST behavior may seriously affect the fatigue life of the panel structure.When the magnitude of the SPL increases from 110 to 130 dB and the thermal load of Δ/Δ ,thin = 2.75 is maintained, the LV (w/h > 2.0) about the flat position of the panel is shown in Figure 8.As shown in Figures 5-8, the different loading conditions of the combined thermal and acoustic loads result in different nonlinear dynamic responses of the thin panel.In addition, in most cases, the stress and displacement response trends are quite similar to each other.

Thick Panel Results
3.4.1.Thermal Postbuckling Analysis.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 9 shows the thermal postbuckling analysis results for the thick panel.As can be seen in the figure, after thermal buckling, the postbuckled deflection of the thick panel increases monotonically.The postbuckling behavior of the thick panel is quite similar to the previous result for the thin panel.Note that the postbuckled deflection is nondimensionalized using the thickness of the thin panel, h thin , and the temperature increase is represented by the critical temperature change for the thin panel (Δ ,thin ).The postbuckled displacement of the thick panel is smaller than that of the thin panel, and the critical buckling temperature change of the thick panel (Δ ,thick = 39.466∘ C) is much higher than that of the thin panel (Δ ,thin = 1.098 ∘ C).

Random Vibration Analysis.
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 (Δ ,thick = 39.466∘ C).Figures 10(a) and 10(b) show the nondimensional maximum displacement response and the stress response at the center of the bottom surface under a random acoustic load at an SPL of 120 dB and without thermal load (Δ/Δ ,thick = 0).As shown in the figure, for the thick panel, SV is observed on the flat position (w/h < 0.2) because the panel structure is not thermally buckled.As can be seen in Figure 10(b), the stress response at the center of the bottom surface is similar to the displacement response given in Figure 10(a).When the combined loads with Δ/Δ ,thick = 2.75 and SPL = 130 dB, which are the same nondimensional thermal load and the same sound pressure level for random acoustic loading, shown previously in Figure 8, are considered for the thin panel, the displacement response in Figure 11(a) shows VBP.random response of the stress is observed by alternating the positive and negative stresses.For the thick panel, the stress response is similar to the displacement response, in the same way as in the previous case for the thin panel (Figure 7). Figure 13 shows the nonlinear dynamic response under combined loading conditions at an acoustic load (SPL = 150 dB) larger than the SPL under the ST condition, while maintaining the nondimensional thermal load (Δ/Δ ,thick = 2.75).As shown in Figure 13, the LV response (w/h > 2) is observed.For the thick panel, the trends of the stress responses are very similar to the shapes of the displacement responses shown in Figures 10-13.

Conclusion
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- time integration method.The nonlinear random responses under the combined thermal and acoustic loads were investigated in the time domain.In the case of the thin panel, the ST response was observed under combined loads with Δ/Δ ,thin = 2.75 and SPL = 110 dB.In the case of the thick panel, the ST response was observed under the loading condition with Δ/Δ ,thick = 2.75 and SPL = 150 dB.In this study, the conditions of combined loads causing the ST behavior were different for the thin and thick panels.In the future, the realistic panel structural model including stiffeners or heat shielding layer will be considered to investigate the ST responses of the panel of high-speed flight vehicles under combined thermal and acoustic loads.

Figure 11 (
b) shows the stress response under Δ/Δ ,thick = 2.75 and SPL = 130 dB.The stress response is similar to the displacement response of the previous cases for the thin panel.Because the panel is buckled in the upward direction as shown in Figure11(a), the compressive stress at the center of the bottom surface is as shown in Figure11(b).However, when a thermal load of Δ/Δ ,thick = 2.75 and an acoustic load of SPL = 150 dB are considered, the ST behaviors observed are as shown in Figure12(a).In Figure12(b), the stress at the center of the bottom surface for the ST response is unique, in comparison to the previous cases shown in Figures10(b) and 11(b) for the thick panel.In the ST response, the

Table 1 :
Natural frequencies of thin and thick square panels (units: Hz).

Table 2 :
Validation of random acoustic load generation.
3.2.Validation of Random Acoustic Load Generation.Before investigating the nonlinear behavior of the panels under combined loads, the generation of a random acoustic load is validated, as presented in Table2.For this validation, p 0 and  u in (