Time-varying stability and vibration of an embedded thin plate on a swept wing
Introduction
In recent years, benefiting from the wide application of various new intelligent materials, morphing aircraft have been extensively developed and attracted more and more researchers' attention. Morphing aircraft can change the aerodynamic shape of wings according to flight conditions to improve fuel economy and aerodynamic performance. In the deformation process, the thin plate structure on the wing interacts with the external fluid to form a complex time-varying system. It is known that the interaction between the thin plate and the outside airflow contributes to the divergence and flutter of the system, which are disadvantageous to the safety and reliability of the aircraft. So it is necessary to research the aeroelasticity of the thin-walled structures on the swept wing.
Ajaj et al. [1] reviewed the development of morphing aircraft and summarized the research status of the morphing aircraft. Li et al. [2] discussed the outstanding case of morphing aircraft and summarized recent progress in aerodynamic analysis methods of morphing aircraft. Zhang et al. [3] studied the nonlinear dynamic behavior of the unfolded wing in subsonic airflow and the dynamic response of the wing in subsonic airflow. Zhang and Wu [4] simplified the physical model of variable span and variable sweep morphing aircraft based on the Kane dynamics method. Guo et al. [5] established the nonlinear partial differential equation of Z-shaped wings in subsonic flow. They studied the nonlinear dynamic response of the morphing wing during deformation. The numerical results were verified by experiments. Yue et al. [6] proposed a sliding mode flight controller and established the nonlinear time-varying motion equation of tailless telescopic wing morphing aircraft. The simulation results showed that the telescopic wing effectively improved the maneuverability of the aircraft. Shalymov et al. [7] proposed the innovative idea of adding rotating “feathers” to aircraft wings to reduce wing vibration. They established a mathematical model of the feather dynamics system and obtained the multi-agent control law. Their research offers a fresh approach to regulating wing flutter.
The dynamic behavior of plate structures has been studied extensively. Hebali et al. [8] investigated the dynamic performance of the visco-Pasternak foundations and discussed the influence of the variable Winkler coefficient, the constant Pasternak coefficient, and the damping coefficient of the elastic medium on the system. Bouzid et al. [9] built a mathematical model of a simply supported FG plate based on Winkler-Pasternak elasticity and examined the dynamic response of the plate in a hot, humid atmosphere. A novel theory of high-order shear and normal deformation was put forth by Kouider et al. [10], [11], and used to explain plate vibration. They compared their theory's precision to other three-dimensional higher-order theories and performed dynamic analysis on functionally graded materials (FGMs) sandwich plates. Furthermore, the stability and dynamics of plates submerged in axial flow have evolved into a classic issue in the study of fluid-structure interaction in recent decades. Many researchers have made in-depth research on this problem. Tang and Dowell [12] studied the limit cycle vibration of a two-dimensional plate under low subsonic flow and discussed the forced nonlinear and aeroelastic responses of the plate by numerical analysis. Tubaldi et al. [13], [14] analyzed a periodically supported rectangular plate immersed in axial flow. The effects of plate aspect ratio and channel height on plate divergence instability and critical vibration velocity were discussed. Yao and Li [15] investigated the nonlinear dynamic behaviors of an embedded plate interacting with an external axial subsonic potential flow. They observed some novel nonlinear mechanical phenomena. Gibbs et al. [16] analyzed the flutter behaviors of cantilever plates in three-dimensional axial flow. They discussed the influences of structural mass ratio and structural aspect ratio on the flutter behavior of cantilever plates. Liu and Yao [17] established the time-varying dynamic model of the device entering the finite air flow field. They concluded that different system parameters would significantly impact on the stability and vibration characteristics of the plate entering the finite air flow field. Zhang et al. [18] studied the dynamics of an inverted cantilever plate with cracks in an axial subsonic environment and obtained the dynamic model of the cracked plate by using function approximation. Lin et al. [19] established a nonlinear model of a composite plate embedded in shape memory alloy (SMA) in a thermal-aero-acoustic coupled field and discussed the nonlinear dynamic response and flutter behavior of the plate. Bulkarev and Lekomtsev [20] analyzed the stability of a rectangular parallel plate subjected to external and internal supersonic fluids. They focused on the effect of the combined action between the two flows on the stability of the plate. Saidi et al. [21] studied the stability and vibration of porous plates reinforced by graphene platelets in supersonic flow, and the influence of different plate parameters on the stability of the plate was discussed. Yao and Qiao [22] used a nonlinear energy sink (NES) and a giant magnetostrictive-piezoelectric energy harvester (NES-GMP) for vibration suppression and energy harvesting in embedded plates exposed to subsonic airflow. They proposed the optimal locations for the NES.
Furthermore, some scholars have conducted extensive research on the dynamic behavior and nonlinear response of the plate in yawed flow. Zhong et al. [23] proposed a way to analyze the stability of magnetic-thermo-elastic functionally graded plates in yawed supersonic flow and discussed the influence of various parameters of the system on the flutter instability of the plate. Khalafi and Fazilati [24] developed an enhanced isogeometric finite element method to analyze the supersonic flutter of a material with variable stiffness in yawed flow and the influence of different parameters on the stability of the plate was discussed. Majidi et al. [25] studied the supersonic flutter of functionally graded carbon nanotube reinforced cantilever trapezoidal cantilever plate in yawed supersonic flow and discussed the influence of various parameters on the flutter boundary of the plate. Gibbs et al. [26], [27] analyzed the stability of a plate during the transition from wing flutter to flag flutter in yawed subsonic flow. Their results showed that the stability boundary of the two states of an airfoil was related to the relative spacing between the first torsion natural frequency and the second bending natural frequency. They conducted aeroelastic experiments in a wind tunnel to verify the validity of the aerodynamic model.
A full theoretical framework for the fluid-structure coupling of plate structures has currently been established. However, the dynamic behavior of submerged plate structures in axial or yawed supersonic flow has been widely examined and discussed, while there are relatively few investigations on the stability and vibration of plate structures submerged in yawed subsonic airflow. The dynamic characteristics of plates in subsonic airflow are different from those in supersonic airflow, which needs further research. Additionally, the yaw angle of the airflow changes when the plate moves in the flow field, causing a complex dynamic time-varying problem for the system's damping and stiffness matrix. It can be predicted that, with the development of material science in recent years, the employment of Morphing aircraft will gradually shift from the supersonic area to the subsonic field. Therefore, the dynamics research of deformable wings under the subsonic environment will be helpful to the design and development of aircraft.
In this paper, a time-varying motion model of a simply supported plate embedded in a swept wing in the subsonic flow is established. The vibration and stability of the thin plate are analyzed. The effect of centrifugal force on the natural frequency of the plate is considered. The influences of flow velocity, rotation speed, rotating shaft distance, length-width ratio, and thickness on the stability and displacement time response of the system are discussed.
Section snippets
Structure modeling
Fig. 1 depicts the wing of a sweep-wing aircraft in two different states. By altering the angle between the wing and the aircraft fuselage, the aerodynamic shape of the aircraft can be modified to accommodate various flight circumstances. A rotating unfolded thin plate in subsonic flow is shown in Fig. 2. The thin plate is embedded in a swept wing with angular velocity Ω. The direction of the counterclockwise rotation is set to positive. The length, width, and thickness of the thin plate are a,
Convergence analysis
The material and geometric parameters of the plate are: a = 1 m, b = 0.4 m, h = 0.001 m, ρ = 7850 kg/m3, Gpa, . The density of the fluid is = 1.29 kg/m3. The harmonic external excitation and damper are actuating on (a/3, b/3) on the plate. The amplitude of external excitation F and the external excitation frequency ω are 1 N and 30 rad/s. The damping of the system is concentrated on the damper and the damping coefficient c is 30 Ns/m.
The distance H, the rotational speed Ω and the
Effectiveness of the plate model
In this section, the dimensionless fundamental frequency of the plate in axial subsonic flow is compared with the simulation results in ANSYS to verify the effectiveness of the plate model.
The linear equation of the plate can be obtained by ignoring the rotating effect nonlinear, the geometric nonlinear term and the external excitation term in Eq. (57).
The solution of Eq. (63) is denoted as , in which is the eigenvector, and ω is the eigenfrequency of the system.
Numerical and simulation and discussions
Fig. 4 (a) depicts the effects of the rotating movement on the fundamental frequency of the plate. It can be shown that the rotation enhances the stability of the plate. The fundamental frequency of the plate rises with the distance H increasing. Fig. 4 (b) displays the variation of the fundamental frequency of the plate in the static flow field. Due to the rotational motion, the relative velocity between the plate and the fluid decreases the fundamental frequency of the plate. Similarly, the
Conclusions
In this paper, the stability and vibration of the embedded plate on the swept wing in the yawed subsonic flow environment are investigated. The main contributions of this study are the development of the potential flow theory-based coupling model of thin plates and yawed subsonic flow and the establishment of the time-varying motion model of swept-wing embedded simply supported plates operating in subsonic flow. By calculating the natural frequency of the plate and the vibration amplitude, the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This research is supported by the Natural Science Foundation of Liaoning (2020-MS-092) and the Natural Science Foundation of China (51975511).
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