Metastructure plate with periodically embedded nonlinear vibration absorbers for nonlinear aeroelastic suppression

A metastructure plate featuring periodically embedded nonlinear vibration absorbers (NVAs) is proposed for the passive suppression of nonlinear aeroelastic responses under supersonic flow conditions. Using the von Karman large deformation theory and supersonic piston theory, the motion equations of a supersonic functionally graded material plate coupled with NVAs are derived from the Hamilton principle. Linear flutter analysis shows that the multiple-NVA design can significantly enhance the aerothermoelastic stability of the metastructure plate. Subsequently, the nonlinear aeroelastic behaviors of the plate and the energy transfer mechanism between it and the NVAs are examined using an energy-based analysis approach. The comparison of bifurcation diagrams indicates that the attachment of periodic NVAs realizes a superior suppression of vibration absorption than a single NVA. Numerical results show that the nonlinear dynamic responses of the plate can be substantially reduced via the targeted energy transfer of NVAs in the post-flutter regime. In particular, the passive control performance of the periodic NVAs does not degrade under an increase in the dynamic pressure. Furthermore, a significant reduction of more than 95% in the response amplitude of the plate can be realized by properly tuning the NVA parameters. The present work demonstrates that


Introduction
In aerospace science and engineering, the aeroelasticity of the plates and shells of high-speed aircraft is a focal research topic [1][2][3]. Because of their structural properties, the plate/shell structures exhibit nonlinear motions under supersonic/hypersonic flow; furthermore, the lightweight structure, high speed, and high manoeuvrability of high-speed aircraft puts them at risk of severe aerodynamic loads. Though they can improve flight performance, such design concepts may induce flutter instability and complex nonlinear dynamic phenomena. Hence, panel flutter and nonlinear aeroelastic suppression have attracted considerable attention amongst researchers, and an increasing numbers of control approaches have been proposed to mitigate or eliminate the adverse impacts of nonlinear phenomena.
Passive control techniques, excellent robustness without external energy input, have been applied for vibration suppression. A nonlinear energy sink (NES) is a passive vibration-control device that can capture and dissipate broadband vibration energy through targeted energy transfer (TET) from primary structures [4][5][6][7].
Compared with the linear tuned mass damper (which is only applied for narrow frequency bands), a well-designed NES can improve the vibration absorption and realize highly efficient energy transmission. NESs have been widely developed and applied to various fields, including architecture [8], machinery [9][10][11], aerospace [12][13], and engineering structures. To improve the control performance of NESs, different NES variants have also been designed and assessed, including bistable [14][15], vibro-impact [16][17], lever-enhanced [18][19][20], and energy-harvesterenhanced [21][22] NESs. Several studies on NES-based panel-flutter suppression have provided valuable insights into the passive-flutter control mechanisms of NESs [23][24][25]. For example, Zhang et al. [23] analyzed the vibration mitigation of a supersonic composite laminated plate, and they obtained superior attenuation under pre-flutter conditions. Pacheco et al. [24] used an energy analysis approach to evaluate the panel-flutter suppression performance of an NES, and they discussed the influence of the NES parameters on its control performance. Subsequently, we applied an NES to the nonlinear aeroelastic suppression of a three-dimensional (3D) hypersonic wing by constructing a 3D wing with an NES-based structure design. The obtained results demonstrated excellent aeroelastic behaviors [25]. Moreover, existing studies have shown that the installation position of the NES plays an important role in its behavior and efficiency, and its performance is also sensitive to parameter variations [7,[25][26]. However, the NES has a limited influence on the dynamic behavior of the system, i.e., the vibration reduction and flutter enhancement.
Metastructures, a metamaterial-inspired concept, feature distributed periodic vibration absorbers [27][28][29]. These structures exhibit interesting dynamic behaviors (e.g., bandgaps and pass bands) that have attracted significant research interest in recent years. The metastructures can behave unique dynamic properties because of the periodic arrangement and parametric design of their local resonators. Although classified as a passive control approach, the ability to suppress multiple modes simultaneously is a key feature of metastructures. The application of local resonators can reduce the structure vibrations by allowing waves of a certain frequency to propagate along the periodic substructure. Furthermore, waves in other frequency bands are unable to pass.
Owing to these superior dynamic behaviors, a number of metastructure studies have focused upon the vibration suppression of systems. Reichl and Inman [30] proposed a conserved-mass 1D metastructure model to suppress the axial vibrations of a host structure. The numerical results indicated that an excellent attenuation was achieved by varying the natural frequency of the local resonators. Ji [31] designed a nonlinear vibration absorber using three-to-one internal resonances, to suppress nonlinear vibrations and eliminate saddle-node bifurcations of the primary structure. Using a distributed array of local vibration absorbers, Casalotti et al. [32] evaluated the multi-mode vibration absorptions of a nonlinear metamaterial beam. The vibration-suppression performance was improved by tuning the parameters of the local absorbers. A type of graded metamaterial beam with attached local resonators was also applied by Hu et al. [33] for broadband-vibration suppression. Basta et al. [34] proposed a mechanical rotating metamaterial cantilever beam coupled to a periodic array of spring-mass-damper subsystems, to realize vibration control. In terms of the designs of the metastructure plates, Sheng et al. [35] analyzed the vibration behaviors of a sandwich plate containing viscoelastic periodic cores; they found that these cores could provide a better attenuation performance than uniform viscoelastic ones. Subsequently, Sheng et al. [36] optimized a lightweight nonlinear acoustic metamaterial beam to realize low-frequency, broadband, high-efficiency vibration reduction. Furthermore, Li et al. [37] designed a sandwich-like plate containing mass-beam resonators for vibration attenuation and isolation, and the performance of this structure was validated theoretically and experimentally. Later that year, Song et al. [38] proposed metamaterial sandwich panels comprising a host sandwich panel and periodically attached resonators, to analyze the vibration and sound properties of the system. The use of the periodic structure design may represent an effective approach for suppressing vibrations, sound radiation, and sound transmission over a wide frequency range.
As demonstrated by the aforementioned studies, various metastructure designs can be used to effectively tune bandgaps and suppress vibrations. Casadei and Bertoldi [39][40] proposed periodic arrays of airfoil-type vibration absorbers to allow wave propagation along a beam, and their design achieved wide band gaps by harnessing fluid-structure interactions and varying the airfoil properties. However, to the best of our knowledge, most studies of metastructures have focused on the vibration mitigation of linear metastructures, particularly for wave propagation and vibro-acoustic suppression. In addition, the design of metastructures with local resonators or periodic substructures has seldom been applied to suppress the vibrations of nonlinear aeroelastic systems. In supersonic/hypersonic aircraft, the skins of wings and engines are exposed to thermal and aerodynamic loads during high-speed flight, which may produce rich nonlinear dynamic phenomena. Therefore, it is essential to investigate how the metastructure design can be applied to the nonlinear aeroelastic suppression of a 3D supersonic plate.
The objective of the present study is to extend the metastructure concept to the nonlinear aeroelastic suppression of a plate. We focus on assessing the effects of a metastructure embedded with periodically distributed nonlinear vibration absorbers (NVAs) on the passive control of flutter stability and the nonlinear aeroelastic response of a functionally graded material (FGM) plate. The remainder of this paper is organized as follows: in Section 2, we formulate the nonlinear aeroelastic equations of the metastructure FGM plate coupled with multiple NVAs; in Section 3.1, the convergence and accuracy of the proposed analytical model are validated; in Sections 3.2 and 3.3, we evaluate the effects of the NVA parameters on the linear flutter and nonlinear aeroelastic behaviors of the metastructure plate, respectively; the main conclusions are drawn in Section 4.

Theoretical analyses
We consider a metastructure that consists of an FGM plate coupled with a distributed array of Na (= na × na) nonlinear absorbers, as shown in Fig. 1. The local NVAs are modelled as small masses mj attached to certain positions (xj, yj) of the plate by a linear damper cj and a spring with linear stiffness coefficient kl,j and nonlinear stiffness coefficient knl,j. The multiple NVAs are periodically and evenly distributed across the plate, with the spaces between the NVAs and plate assumed sufficiently large to prevent the NVAs from colliding with the plate during motion. Thus, new displacements pj (j = 1,2, …, Na) must be introduced as additional degrees of freedom (DOFs).

Material properties
Because of their excellent thermo-mechanical performances, FGMs have potential applications in aerospace engineering structures [41][42]. Therefore, the nonlinear aeroelastic behaviors of a supersonic FGM plate are here examined. In general, an where E, ν, ρ, α, and κ denote the elastic modulus, Poisson's ratio, density, thermal expansion coefficient, and thermal conductivity, respectively. The symbol p denotes the volume-fraction index. The superscripts u and b denote the upper and bottom surfaces of the FGM plate, respectively.
The temperature-dependent material properties can be written as ( ) where P0, P-1, P1, P2, and P3 are coefficients of temperature T(K).
To consider the thermal effects, we assume that the temperature varies only in the thickness direction and that the in-plane temperature field remains constant. Thus, the temperature-field distribution throughout the plate can be obtained using the 1D Fourier heat-conduction equation, expressed as where κ(z) is the thermal conductivity of the FGM plate. The upper and bottom temperature boundary conditions are ( 2) The solution of Eq. (3) can be obtained as The temperature change along the thickness is defined as To simulate aircraft structures under the thermal load conditions that occur during supersonic flight, the temperature T b exerted on the bottom surface is defined as the reference temperature T0, and the upper surface is exposed to a high uniform temperature field.

Energy expressions
Using the first-order shear deformation theory, the displacement components for the FGM plate in the x-, y-, and z-directions can be written as In Eq. (6), the strain vector can be divided into linear and nonlinear components, as where T The constitutive relation of the FGM plate considering thermal effects can be written as where σ and α denote the stress and thermal expansion coefficient vectors, respectively. Here, C denotes the elastic constant matrix. The vectors and matrices are given as follows:   Substituting Eq. (4) into Eq. (12), the kinetic energy can be written as where ( ) The potential energy Up of the metastructure plate under thermal loads is given as The Rayleigh's dissipation function of the nonlinear resonators is The external work performed by the aerodynamic pressure can be obtained using and the aerodynamic pressure Δp is modelled using first-order piston aerodynamics, expressed as where q∞  is defined as the plate stiffness.
ρm0 and Em0 denote the density and elastic modulus of the metallic material, respectively.
The total energy function of the metastructure plate can be obtained as

Governing equations in state-space form
The displacement components of the FGM plate are expressed as where For a simply supported FGM plate, the shape functions can be expressed as According to the Hamilton principle, the governing equations for the complete system can be obtained as

Energy evaluation
To explore the energy transfer of the proposed metastructure plate and its passive control performance, several quantities (e.g., the total mechanical energy contained in the plate, the energy input by the airflow, and the energy dissipated by the NVAs) are estimated using an energy-based analysis approach.
The total instantaneous energy of the plate is expressed as The total instantaneous energy of the NVAs is expressed as The energy input by the airflow and dissipated by the NVAs for t ∈ [0, T] are obtained as respectively.

Results and discussions
In the present work, the passive flutter control and nonlinear aeroelastic behaviors of a supersonic FGM plate featuring periodically attached NVAs are assessed. The FGM plate was composed of Si3N4 and SUS304, and its temperature-dependent material properties are listed in Table 1. The mass density and thermal conductivity of Si3N4 are ρ u = 2370 kg/m 3 and κ u = 9.19 W/(mK), respectively; those of SUS304 are ρ b = 8166 kg/m 3 and κ b = 12.04W/(mK), respectively. The Poisson's ratio is defined as ν u = ν b = 0.28, and the shear correction factor is defined as κs = 0.91 [43]. For all case studies, the parameters were set as a = b = 1.0 m, h = 0.02 m, and μ/Ma = 0.01.
In numerical simulations, all figures were plotted with respect to a typical point (x, y) = (0.75a, 0.5b) of the plate. Table 1. Temperature-dependent material properties of Si 3 N 4 and SUS304. Materials

Convergence study and validation
The convergence and accuracy of the proposed formulation were validated for the  Xie et al. [44]. The comparison is shown in Fig. 4 and indicates strong consistency.
Consequently, the aforementioned results demonstrate that the proposed analytical model predicts the aeroelastic stability and nonlinear dynamic responses of the metastructure plate with sufficient accuracy.

Linear flutter analysis
We   For larger numbers of NVAs, a significant improvement in the aeroelastic behavior of the metastructure plate was observed. As shown in Fig. 6, the plates coupled with 3 × 3, 4 × 4, 5 × 5, and 6 × 6 NVAs exhibited an enhanced aeroelastic stability over the entire parameter domain. Considering the case of 6 × 6 NVAs for kl = 20 as an example (Fig. 9), we see that the flutter stability of the metastructure was enhanced by 16.46%, and the NVA mode became unstable at λcr = 764.87. As kl was increased further, the flutter boundary of the system finally stabilized near λ = 655.3.

Furthermore, a similar trend was observed for the cases of 3 × 3, 4 × 4, and 5 × 5
NVAs. Therefore, multiple attached NVAs can enhance the aeroelastic stability of the supersonic FGM plate.

Nonlinear aeroelastic analysis
In this section, the geometrical nonlinearity of the plate and the nonlinear stiffness of the NVAs are considered. The effects of the distributed NVAs on the nonlinear aeroelastic behaviors of the metastructure plate were carefully analyzed. Meanwhile, an energy-based approach was used to evaluate the energy transfer of the present metastructure. Fig. 15 shows the bifurcation diagrams of the local amplitude extremes for the pure and metastructure plates. Here, the parameters of the NVAs were set as mj = 0.01, cj = 0.1, kl,j = 10, and knl,j = 2000. As shown in Fig. 15  When 742 < λ < 798, the pure plate exhibited a stable LCO; however, because of the attachment of 6 × 6 NVAs, the motion of the metastructure plate converged to a static equilibrium. Fig. 16(a) compares the nonlinear aeroelastic responses at λ = 780 for the pure and metastructure plates, and it confirms that the addition of periodic NVAs can improve the aeroelastic stability of the system. In addition, the position of the 28 th NVA (x28 = 3/4, y28 = 7/12) was close to the typical point; hence, its response was compared with that of the plate, as shown in Fig. 16(b). The vibration amplitude of this NVA was much smaller than that of the plate, which indicates that no targeted energy transfer occurred. The system's own stability in the pre-flutter regime means that the vibration energy of the plate is insufficiently strong to meet the energy threshold of the TET. Thus, the absorption energy of NVAs is small. To better understand the energy transfer mechanism of NVAs, the time histories of the energy for the present system are depicted in Fig. 16(c). As time elapsed, the input and dissipated energy gradually increased and eventually converged. During this process, the input energy was entirely dissipated by the NVAs. Accordingly, the remained energy in the system was reduced to zero.  Fig. 17(b), the vibration amplitude of the NVA was still smaller than that of the plate. The corresponding energy analysis is shown in Fig.   17(c). The energy dissipation and energy input are clearly seen to reach a balance, and the remained energy in the system maintains the stable LCO of the plate.
Furthermore, the time histories of the plate energy and the total energy of the NVAs are shown in Fig. 17(d), where we can see that the total NVA energy is far below that of the plate and the TET phenomenon does not occur. Under a further increase in λ, the plate produces a random response with a small amplitude. The case of λ < 900 is shown in Fig. 18. Similarly, the plate's response was reduced by ~85%. Nevertheless, the metastructure plate exhibited a recurrent motion with an undulating amplitude, and the amplitude of the 28 th NVA exceeded that of the plate. The energy time histories shown in Fig. 18(c) indicate a dynamic balance of energy input and dissipation. To better understand the energy transfer mechanism, Fig. 18(d) presents the time histories of the plate energy and the total energy of the NVAs for t ∈ [0, 5].
When the vibration energy of the plate reached the critical energy threshold, the NVA energy suddenly increased, a large quantity of energy was transferred from the plate to the NVAs, and the response amplitude of the plate began to decrease. Thus, the vibration energy of the plate gradually weakened, and the NVAs lost their ability to capture energy. Subsequently, owing to the continuous energy input of the airflow, the energy of the plate increased rapidly again. The aforementioned process was repeated to illustrate the phenomenon shown in Fig. 18(b). Furthermore, this process indicates that a TET was accomplished via recurrent transient resonance capture. In addition, Fig. 19 compares the power spectrum diagrams of the responses for the plate and 28 th NVA. The two peak frequencies of the responses were remarkably consistent, and this NVA had a wider frequency band than the plate. Based on the above analysis, we observe from

Conclusions
In this study, the concept of a metastructure design featuring periodically 3) A significant suppression performance enhancement can be achieved by appropriately tuning the NVA nonlinear parameter, and a reduction of up to 95% was achieved for the response amplitude of the metasurface plate. To summarise, the present study demonstrates the excellent potential of the proposed metastructure for realising performance enhancements in term of both flutter stability and nonlinear dynamic behaviors.
, d The expressions in the matrices M A , D S , D A , D C , K S , K A and K C are expressed as:     T  T  T  T  TT  11  12   T  0 0  T  T  66   1  dd  2 T  T  T  T  T  11  12   T  T  T  T  T  T  66  66   T  T  T  12   22   22   2 T  22   T  T  T  T  T  T  66  66   T  T  T  T  T  T  11  12   2   22 22 T  T  T  T  T  66  66   T  T  T  T  T  T  12  22   TT  T  66   22   22 2 T  T  T  66   T  T  T  T  T  T  11  12   TT   T  12  22 2B TT   T   TT  TT  T  T  66  66   2  2 y y y y -34)   T  T  T  T  TT  11 12 -35)  T  T  T  T  T  T  12  22   T  T  00  T  66   1  dd  2     Comparison of LCO behaviors obtained under the present formulation for an aluminium rectangular plate with a simply supported boundary.

Figure 5
Eigenvalue solutions of the rst ve modes (under non-dimensional dynamic pressure λ) for the FGM plate (p = 1.0) without NVAs.

Figure 6
Critical utter aerodynamic pressure versus NVA linear stiffness kl for different number of NVAs.     Flutter behaviors of the metastructure plate with 6 × 6 NVAs for NVA linear stiffness coe cients of kl = 20.

Figure 10
Flutter behaviors of the metastructure plate with 6 × 6 NVAs for different volume fraction indices: (a) critical utter aerodynamic pressure and (b) utter frequency.

Figure 13
Effects of temperature rise on the utter stability of the metastructure plate with 6 × 6 NVAs: (a) p = 0.5 and (b) p = 2.0.      Power spectrum diagrams of the responses for the plate and 28th NVA at λ = 900.

Figure 20
Bifurcation diagram of the metastructure plate with different NVA nonlinear stiffness coe cients knl,j at λ = 1000.

Figure 21
Bifurcation diagram of local amplitude extreme for the improved metastructure plate (knl,j = 40000).