Abstract
Advanced aerospace structures are subjected to extremely harsh working environment, including of mechanical, acoustic, thermal, and aerodynamic loads. These combined loads can make the structures exhibit complex response, which has already attracted increasing concern in the design of advanced aerospace structures. To tackle the problem, a finite element model (FEM) by discretizing the theoretical differential equation is established to calculate the dynamic response of thinwalled structures under combined thermal and acoustic loads. The numerical analysis indicates that three types of nonlinear responses of the thin plate under combined thermalacoustic loads are obtained, and the mechanism of snapthrough is revealed through two ways: thermal buckling analysis and thermal modal analysis. which can be used to explain the nonlinear response characteristics.
Highlights
 FEM model by discretizing the theoretical differential equation is established to calculate the dynamic response of thinwalled structures under combined thermal and acoustic loads.
 Three types of nonlinear snapthrough response of thinwalled structures under thermalacoustic loads are obtained.
 The mechanism of snapthrough is revealed through two ways: thermal buckling analysis and thermal modal analysis.
1. Introduction
Aerospace vehicles widely adopt thinwalled structures to reduce structural weight, such as missile wings, warhead casings, aeroengine combustor flame tube and afterburner liner, etc. These structures bear complex combination of mechanical, acoustic, thermal, and aerodynamic loads during operation. The impact of combined loads on the fatigue life of thinwalled structures is considerably more severe than the simple superposition of individual loads. These combined loads may lead to nonlinear structural response characteristics [1]. Specifically, thinwalled structures are prone to snapthrough response under the combined thermalacoustic loads, and the reverse stress generated by the snapthrough response can reduce the structure's fatigue life dramatically [2]. Therefore, it is necessary to probe into the nonlinear response of thinwalled structures under thermalacoustic loads.
Thinwalled structures first experience thermal buckling under the increasing thermal load, and the resulting geometric nonlinearity increases the structural stiffness. Then, the post buckling structures will exhibit different response characteristics excited by different levels of sound pressure. Ng and their research group first observed the phenomenon of snapthrough in thinwalled plates in the thermal buckling state under strong noise conditions. Subsequently, they proposed the use of a single degreeoffreedom (SDOF) model to analyze the statistical response characteristics of thinwalled plates under thermalacoustic loads, and pointed out the drift of resonance frequency and the broadening of resonance band in nonlinear response [3]. Jon Lee conducted extensive research on the statistical responses of displacement and strain by using the SDOF model, indicating that displacement and strain follow the FokkerPlanck distribution [4]. Mei has researched buckling and dynamic response problems in aerospace structures under thermalacoustic loads by using FEM method for three decades, which focused on the application of iterative method in response solving [5]. Sha et al. have conducted extensive simulations on the thermalacoustic load responses of thinwalled structures and also discovered the snapthrough response [6, 7]. The above researches focus on the analysis of structural dynamic response under thermalacoustic loads, while there is relatively little research on the modal characteristics of thermal structures and their impact on the mechanism of snapthrough response.
The objective of this paper is to predict the nonlinear snapthrough response of thinwalled structures under thermalacoustic loads and reveal its mechanism. A twoedge fixedsupported rectangular plate is selected as the modelling object. The thermal load is simplified as constant throughout the plate, and the acoustic load is simulated using band limited Gaussian white noise according to a modified triangle series method. The FEM model of the plate by discretizing the theoretical large deflection differential equation is established to simulate the nonlinear response under thermalacoustic loads. The mechanism of nonlinear snapthrough response is researched through two ways: thermal buckling analysis and thermal modal analysis. The findings provide valuable insights for the design of structures operating in hightemperature environments, especially in structural antiacoustic fatigue design.
2. Theoretical model of thermalacoustic vibration for thin plate
2.1. Governing equations of thin plate under thermalacoustic loads
The wellknown von Karman large deflection equation with thermal and acoustic loads can be derived by the analysis of forces and moments in the plate:
where $D$ is bending stiffness, $h$ is the thickness of the plate, $\mu $ is Poisson ratio, $\theta $ is the temperature gradient along $h$, $p\left(x,y,t\right)$ is the random pressure used to simulate acoustic load, $\mathrm{\Phi}$ is Airy stress function, $w$ is transverse deflection of the middle plane.
The corresponding strain compatibility equation can be represented as:
where $E$ is Young’s modulus, ${\nabla}^{4}$ is biharmonic operator, $\stackrel{}{T}\left(x,y\right)$ is average temperature along the plate thickness $h$.
2.2. FEM equation of thin plate under thermalacoustic loads
It’s very difficult to directly solve Eq. (1) and Eq. (2), which can be discretized into dynamic FEM equation as:
where, $\mathbf{M}$ is the mass matrix, $\mathbf{C}$ is the damping matrix, ${\mathbf{K}}_{L}$ is the linear stiffness matrix, ${\mathbf{N}}_{1}$ and ${\mathbf{N}}_{2}$ are firstorder and secondorder nonlinear stiffness matrices respectively, $\mathbf{X}$ is the displacement vector, ${\mathbf{F}}_{p}$ is the acoustic load vector, The influence of thermal load is reflected on both sides of Eq. (3), i.e. ${\mathbf{K}}_{T}$ represents the stiffness term induced by thermal stress, and ${\mathbf{F}}_{T}$ is the thermal load vector.
The above stiffness terms can be combined into one term, and the same applies to the load terms, so Eq. (3) takes the simplified form as:
By employing the NewmarkBeta integration method, the dynamic response of the thin plate under the combined thermalacoustic loads can be simulated.
2.3. Determination of thermalacoustic loads
This paper assumes that the thermal load on the thin plate is a uniformly distributed temperature field, and there is no temperature gradient considered. The buckling coefficient can be prescribed as:
where ${T}_{ref}$ is the ambient temperature, ${T}_{cr}$ is the critical buckling temperature.
The acoustic load is a bandlimited Gaussian white noise excitation, with its amplitude following the zeromean Gaussian distribution and its power spectral density (PSD) that is uniformly distribution. The effective value of the sound pressure is:
The PSD can be given by:
where $\mathrm{\Delta}\omega ={\omega}_{u}{\omega}_{l}$ is bandwidth represented by angular frequency.
3. Numerical example
3.1. Analysis model
This paper adopts the sequentially coupling method to address the thermalacoustic vibration problem. A typical aluminum alloy thin plate is selected as the research model, which had clamped boundary conditions at both short edges. The length, width, and height of the plate are 457 mm, 25 mm and 2.3 mm, respectively. The material parameters are Young’s modulus $E=$7.3×10^{10} Pa, density $\rho =$2763 kg/m³, Poisson’s ratio $\mu =$0.3, and linear expansion coefficient $\alpha =$22.3×10^{6 }°C.
3.2. Thermal buckling analysis of thin plate
Using Galerkin’s approach, the large deflection equation Eq. (1) can be reduced to a secondorder ordinary differential equation in modal coordinates. In order to illustrate the basic characteristic and explain the mechanism of snapthrough, the nondimensional singlemode plate equation can be derived as:
where, $q$ is the generalized coordinate of displacement in $w(x,y,t)=q\left(t\right)h\mathrm{s}\mathrm{i}\mathrm{n}\pi x\mathrm{s}\mathrm{i}\mathrm{n}\pi y$, $k=4A$, $A=\frac{3}{4}\left[\right(1{\mu}^{2}\left)\right(1+{\beta}^{4})+2({\beta}^{4}+2\mu {\beta}^{2}+1\left)\right]$, $\beta $ is the aspect ratio of the plate, i.e. the ratio of length to width of the plate, $\xi $ is the damping ratio, and ${p}^{d}\left(t\right)$ is the acoustic load.
The potential energy of Eq. (8) can be given by:
When $\mu =\text{0.3}$, $\beta =\text{1.4}$, and $s=$1, 2, 4, 6, the potential energy changes as shown in Fig. 1. According to Eq. (9) and Fig. 1, when $0<s<1$, the potential energy curve has a single minimum position, indicating that the thin plate has only one stable equilibrium position. However, when $s>1$, the thin plate enters thermal buckling state, and the potential energy curve has two lowest positions, implying that the thin plate has two stable equilibrium positions. These two equilibrium positions are symmetric about the initial geometrical configuration. This means that when the SPL of external acoustic load excitation is small, the thin plate may keep a small vibration near one of the two equilibrium positions. As the SPL of acoustic load increases to a certain level, vibration of the thin plate would shuttle back and forth between the two equilibrium positions, resulting in snap through phenomenon.
When the SPL exceeds the level required for snapthrough, but is not too high, there are some random and intermittent short time periods due to the randomness of the acoustic load. At this time, the SPL is not sufficient to drive the vibration of the plate to jump to another equilibrium position without snapthrough happened, which means the plate can only vibrate at one equilibrium position with small amplitude. Conversely, there are also some random and intermittent short time periods, during which the SPL has ability to drive the large amplitude vibration of the plate to jump to the other equilibrium position with snapthrough happened. We call the above vibration as intermittent snapthrough.
If the SPL is high enough, highintensity acoustic load has sufficient ability to drive the plate to vibrate significantly between two equilibrium positions at any time. In this case, the vibration response is called continuous snapthrough.
Fig. 1Potential energy
3.3. Thermal modal analysis of thin plate
In prebuckling state of the plate, the inplane thermal stress caused by the temperature rise decreases the stiffness of the structure. Thermal buckling occurs at the critical buckling temperature ${T}_{cr}$, which leads to a decrease in structural stiffness to zero. Then, as the temperature continues to rise, unrecoverable buckling deflection will be arised, and the plate will enter postbuckling state. In postbuckling stage, the structural stiffness will increase to a certain extent.
The critical buckling temperature of the thin plate has been calculated to be ${T}_{cr}=$3.76 °C with setting the reference temperature ${T}_{ref}=$0 °C. The first 3 modal shapes for prebuckling with $s=$0, postbuckling with $s=$5.45 and 19.07 are shown in Fig. 2. The modal shapes are significantly different between prebuckling and postbuckling, especially in higher order modal shapes at more serious postbuckling state. The 3rd modal shape has become a bendingtorsion coupling mode, which means higher order modes would impact the dynamic thermalacoustic response.
The first 10 modal frequencies are shown in Fig. 3. In general, the modal frequencies decrease with the increasing temperature before thermal buckling, and reach their minimum values at critical buckling temperature (the 1st frequency is 0). After thermal buckling, due to the influence of thermal hardening effect, the frequencies increase with the temperature further rising. Moreover, the modal frequencies tend to approach each other as the temperature rising in postbuckling, which implies resonance would be induced at the concentrative frequencies and the dynamic snapthrough response may be occurred.
Fig. 2First three modal shapes of thin plates under different buckling coefficients
Fig. 3Changes in the first ten modal frequencies of the thin plate
3.4. Thermalacoustic vibration response of thin plate
In simulated condition settings, taking the reference temperature and the structural temperature as 0 °C and 19.44 °C respectively, the structural temperature, corresponding $s=$5.17, is loaded linearly on the plate within 0.3 seconds, and the acoustic load is loaded on the plate in the whole simulated time period of 1.224 seconds. Dynamic responses of the thermal plate are obtained under SPLs of 149 dB, 152 dB, 158 dB, and 170 dB in the time domain. The transverse displacements of the midpoint in plate under different SPLs are shown in Fig. 4.
Fig. 4Time history displacements of the thin plate under different sound pressures
It can be seen that the vibration of the plate takes place only around a single equilibrium position with the SPL of 149 dB and there is no snapthrough. When the SPL increases to 152 dB, the plate not only vibrates slightly around each equilibrium position, but also shuttles back and forth intermittently between the two balanced positions. The intermittent snapthrough response occurs in this case, resulting in the reversed stress that weakens the fatigue life of the structure. When the SPL rises to 158 dB, although it’s still intermittent snapthrough at this time, the frequency of snapthrough significantly increases. At last, a continuous snapthrough response occurs when the SPL is up to 170 dB.
4. Conclusions
This paper simulates and analyzes the nonlinear snapthrough response of thinwalled structures under thermalacoustic loads and reveals its mechanism innovatively through two approaches: thermal buckling analysis and thermal modal analysis. The main conclusions are as follows:
1) In prebuckling state, there is a single vibration equilibrium position in thin plate. In postbuckling state, there exist two vibration equilibrium positions.
2) Three types of nonlinear responses are obtained: a small amplitude vibration around one of the two equilibrium positions, an intermittent snapthrough shuttled back and forth between two equilibrium positions, a continuous snapthrough response at any time between two equilibrium positions.
3) Modal frequencies decrease with the increasing temperature in prebuckling, and reach minimum values at critical buckling temperature (the 1st frequency is 0). Due to the influence of thermal hardening effect, the frequencies increase with the temperature further rising inpost buckling.
4) The mechanism of snapthrough is revealed through two ways: by thermal buckling analysis, two potential energy equilibrium positions are obtained and by thermal modal analysis, modal frequencies tend to approach each other.
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About this article
The authors have not disclosed any funding.
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
The authors declare that they have no conflict of interest.