Research paperSubcritical and supercritical nonlinear aeroelastic behavior of a morphing wing with bilinear hinge stiffness
Introduction
As a new flight concept, morphing aircraft can adapt to missions that to date have been performed by different types of aircraft [1], [2], [3], [4], [5]. A typical morphing aircraft is the folding-wing aircraft, which by folding its wings during flight can change its configuration autonomously to match different flying environments while still retaining optimum flight performance. However, such in-flight changes in the wingspan and the wing planform area may induce some aeroelastic problems because the structural characteristics definitely change during the morphing process. Furthermore, the inevitable structural nonlinearity in the hinge joints of the folding wings must cause a particular nonlinear aeroelastic behavior.
There have been many previous studies of folding-wing aircraft. As early as 2005, Lee et al. [6] used numerical simulation to study how the stiffnesses of the inboard and outboard hinges affect the flutter characteristics of a folding wing. Matthew et al. [7] used the finite-element (FE) method to investigate how the folding angle, hinge stiffness, and weight affect the natural frequency and flutter speed of a folding-wing aircraft, and they obtained the mode shapes of the aircraft. Tang [8] established a dynamic equation for a folding wing to obtain the first six natural frequencies and mode shapes of the folding wing. Fujita et al. [9] used a multi-body dynamics simulation program to study the dynamic behavior of a folding wing during deformation, and they analyzed quantitatively how the system stiffness, damping, and aerodynamic forces affected the aircraft deformation. Zhao and Hu [10] proposed a parameterized aeroelastic modeling methodology and improved the efficiency of the flutter analysis of a folding wing with different folding angles.
However, the aforementioned studies focused on morphing structures with linear stiffness and ignored the possible nonlinearity such as free-play or geometric nonlinearity [11], which is a common problem in the aerospace engineering and yet to be solved. In the field of nonlinear aeroelasticity, Vasconcellos [12] investigated the effects of a multi-segmented nonlinearity in the pitch degree of freedom on the behavior of a two-dimensional aeroelastic system and found that four sudden transitions take place with a change in the system's response. In fact, there have been several studies of folding wings with nonlinear hinges. Lee et al. [13] conducted nonlinear aeroelastic studies of a folding wing with free-play nonlinearity, using the fictitious mass (FM) method [14] to deal with the nonlinear aeroelastic problems in the same fictitious modal coordinates. However, this method cannot be used directly to analyze how stiffness or other parameters affect a folding wing with free-play or bilinear stiffness in its hinge joints, the reason being that FM-based FE modeling must be repeated each time any parameter is changed. So, in order to reduce the workload, parameterized modeling provides a novel way to solve such problems where otherwise repeated modeling would be necessary. In the field of nonlinear parametric modeling, Huang [15] has proposed a parameterized fictitious mode (PFM) approach for generating the fictitious modal coordinates and representing the structural dynamics efficiently. However, the region of the nonlinear motion and the relationship between the motion and the stiffness, folding angle, or other parameters are yet to be found and defined. This affords an excellent opportunity to investigate the relationship between the structural parameters and the nonlinear aeroelastic response of the morphing wing.
The aim of the present study is to investigate how hinge stiffness affects the structural nonlinear motion of a folding wing so that the best wing configuration can be determined while also considering the nonlinearity. Moreover, to reduce the amount of calculation, parameterized modeling and the PFM method are applied to calculate the aeroelastic behavior of the morphing wing at different folding angles and stiffnesses. This paper is organized as follows. In Section 2 and 3, the parameterized FE model and nonlinear aeroelastic model are presented, respectively. In Section 4, numerical examples are used to analyze in detail the nonlinear response calculated using the proposed parametric model. Finally, some conclusions are drawn in Section 5.
Section snippets
Nonlinear structural dynamic modeling methodology
The structural parameters of the folding wing not only change constantly but are also complex, when the structural nonlinearity, as shown in Fig. 1, is considered. To solve this type of problem, it is necessary to establish a parameterized aeroelastic model. To calculate and analyze the nonlinear behavior efficiently, an efficient nonlinear calculation method also plays a decisive role. Mode superposition is one of the most commonly used methods in dynamic analysis, but modes cannot be
Nonlinear aeroelastic modeling
Direct simulation can also be used to deal with the nonlinear responses of the folding wing in the fictitious modal space. The basic assumption of generalized direct simulation is that the nonlinearity can be measured by a set of system parameters, referred to as the nonlinear parameters. The displacement, velocity or acceleration of nodes, the force acting on the structure all can be defined as nonlinear parameters. Herein, the rotation angles of two hinges are defined as nonlinear parameters.
Numerical simulations and discussion
To investigate how the hinge stiffnesses affect the nonlinear oscillations of the folding wing, it is necessary to select several typical nonlinear configurations to model the nonlinear system, carry out time-domain marching, and then compare and analyze the results. Based on the established parameterized model, the flutter boundary of the folding wing with linear hinge stiffnesses is analyzed first, and some stiffness configurations are selected as the foundation for subsequent research.
Conclusions
Based on parametric modeling and the PFM method, nonlinear aeroelastic modeling was carried out for a folding wing with nonlinear hinges, and how the hinge stiffnesses influenced the nonlinear oscillations of the wing was analyzed. The conclusions are as follows.
- 1)
For a folding-wing aircraft, no matter its hinge stiffness or whether nonlinearity is considered, its aeroelastic characteristics for 0° folding angle are better than those at other folding angles. The configurations with folding angles
CRediT authorship contribution statement
Xinghua Zhou: Software, Writing – original draft. Rui Huang: Methodology, Writing – review & editing, Supervision.
Declaration of Competing Interest
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, ‘Subcritical and supercritical nonlinear aeroelastic behavior of a morphing wing with bilinear hinge stiffness’.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant 11972180 and 12022203, in part by the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) (Grant no. MCMS-I-0121G02). The valuable discussions with X. J. Yao regarding the finite element modeling of the illustrative example are greatly appreciated.
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