Study on the Aeroelastic Characteristics of a Large-Span Joined-Wing Solar-Powered UAV

Abstract

When a joined-wing configuration is applied to the design of solar-powered UAVs, the increasing span amplifies aeroelastic effects, while structure complexity poses greater challenges to computational effectiveness during the conceptual design phase. This paper focuses on a large-span joined-wing solar-powered UAV (LJS-UAV) engineering prototype. The structural finite element model of the whole system is constructed by developing the ‘Simplified beam-shell model’ (SBSM) and verified by a structural mode test. A numerical simulation approach is employed to comprehensively analyse and summarise the aeroelastic characteristics of the LJS-UAV from the perspectives of static aeroelasticity, flutter, and gust response. The mode test identified 30 global modes with natural frequencies below 10 Hz, indicating that the LJS-UAV possesses an exceptionally flexible structure and exhibits highly complex aeroelastic characteristics. The simulation results reveal that the structural elasticity induces significant variations in aerodynamic forces, moments, and derivatives during flight, which cannot be neglected. The longitudinal trim strategies can considerably influence the aeroelastic boundary of the LJS-UAV. Utilising the front-wing control surfaces for trim is beneficial in improving structural performance and expanding the flight envelope.

1. Introduction

High-altitude long-endurance solar-powered UAVs have remained a subject of research interest in recent years due to their advantages in endurance and range. Aeroelasticity has become particularly critical with the deepening of theoretical investigations and accumulated engineering practices. The pursuit of extended flight durations and enhanced payload capacities necessitates an increasingly larger aircraft span, consequently exacerbating aeroelastic challenges. The joined-wing configuration has attracted significant scholarly attention due to its excellent structural performance. This paper aims to provide researchers with novel perspectives for further exploration by investigating the aeroelastic characteristics of the LJS-UAV engineering prototype.

Wolkovitch proposed the concept of joined-wing configuration in the mid-1970s and introduced it in detail in 1986 [1] based on finite element analysis and wind tunnel tests, pointing out the many advantages of the joined-wing configuration. Numerous scholars have since carried out a lot of research [2,3,4,5,6,7,8,9,10] on this configuration and envisioned its application to the next generation of civil aircraft, military transport aircraft, and high-altitude long-endurance UAVs. Karim [11] outlined the main applications of box-wing configuration in transport aircraft, studying the application of this configuration in various types of aircraft, and highlighted the specific performance and functional potential of box-wing configuration. The wings of a joined-wing aircraft are interconnected and form a unique triangular frame structure with the fuselage, exhibiting good stiffness characteristics. It can reduce induced drag by increasing the aspect ratio, thereby improving the lift-to-drag ratio. With consistent lift-to-drag characteristics, the wing span can be reduced, resulting in higher aerodynamic efficiency in the span direction. In addition, the area of the rear wing of a joined-wing aircraft is much larger than the area of the horizontal tail of a conventional layout aircraft, resulting in higher trim efficiency and lower trim drag. The joined-wing configuration has become a promising and competitive configuration due to its multifaceted advantages in aerodynamics, structural mechanics, flight mechanics, etc. However, the design of aircraft with this configuration has enormous challenges and difficulties due to the tightly coupled multidisciplinary interactions between structural dynamics, aerodynamic load distribution, controllability, and stability. Much of the current research remains at the conceptual stage, or the design of small-span aircraft for technical verification, and few aircraft in service have joined-wing configurations, such as High Altitude Performance Demonstrator in Italy [12], and the Chinese ‘Xiaolong’, ‘Wuzhen 7’, etc.

Researchers have carried out a significant amount of work to explore the structural characteristics of aircraft with a joined-wing configuration, such as the discussion of the nonlinear static characteristics of the joined-wing configuration by Marisarla [13] and the structural optimisation design work for a high-altitude long-endurance joined-wing aircraft carried out by He [14]. In 2016, Cavallaro [15] presented and critically examined previous decades of research on joined wings, pointing out how the design of a joined wing can be exacerbated by enhanced intrinsic interactions between different disciplines and the over-constrained nature of the configuration, and the aeroelastic response can show some extremely complex phenomena for flexible aircrafts.

Research on the aeroelastic performance of joined-wing configurations has continued in recent years. Genetic algorithms based on engineering beam theory were used by Li [16] to optimise the beam profile parameters of the joined wing under aeroelastic constraints, achieving structural weight reduction. A coupled model of the stiffness and inertia for a joined-wing aircraft was established by Su [7] through non-linear methodologies to discuss the effect of the lifting surface interaction of a joined-wing aircraft on its aeroelastic performance and confirm that the joined-wing configuration has very rich and complex aeroelastic characteristics. The flutter characteristics of a joined-wing UAV were analysed by Qi [17] through the application of the mode synthesis method. This investigation revealed that a wide range of flutter speeds and frequencies is inherent to joined-wing configurations, and the fuselage stiffness will have an important effect on the flutter characteristics of a joined-wing UAV. The aeroelastic characteristics of the joined-wing configuration and the conventional configuration were compared by Sun [18] through structural design and simulation of a joined-wing aircraft, wishing to highlight the many advantages inherent to the joined-wing design. An effective method for modelling a flexible joined wing was proposed by Liang [19] by transforming the aeroelastic model of the joined wing into a set of differential equations to perform a static aeroelastic analysis of a joined-wing aircraft under thrust within the flutter boundary. The dynamic aeroelastic behaviour of three different configurations of joined-wing aircraft was investigated by Cavallaro [20] using time-domain and frequency-domain methods. Cavallaro [21] studied the dynamic aeroelastic properties of Prandtlplane Joined Wing, performed a detailed analysis of the energy transfer between the fluid and the structure, and investigated in depth the mechanisms that cause aeroelastic instability. Mayuresh [22] studied the non-linear structure and aeroelastic behaviour of joined wings, confirming that the non-plane joined wing structure is superior to the plane joined wing over the conventional wing. Ground experimental methods were used by Bi [9] and Alanbay [10] to explore the performance of the joined wing. Flight tests were conducted by Galinski [23], Richards [24] and Garnand-Royo [25] to explore the aeroelastic response characteristics and flight performance of joined-wing aircraft.

The joined-wing UAV discussed in this paper belongs to the category of high-aspect-ratio aircraft. In research addressing the aeroelastic issues of high-aspect-ratio aircraft, aeroelastic modelling constitutes a core technology. This modelling is divided into three aspects: structural modelling, aerodynamic modelling, and fluid–structure coupling. A highly prevalent approach in structural modelling research is the equivalent beam model, where researchers have developed extensive theoretical frameworks [26,27,28,29,30,31]. However, studies [32] indicate that the calculation of bending and torsional stiffness within beam models is dependent on the choice of constitutive equations, which may lead to discrepancies of up to 40% in aeroelastic results. Obviously, an appropriate description of complex structural kinematics is crucial for predicting aeroelastic instability. Refs. [33,34] indicate that the finite element method is the most reliable approach in structural modelling, wherein the structural model may be described using different element types such as beam elements and shell elements. Numerous researchers have developed beam-shell models based on the finite element method [35,36], which have widely been used in engineering practice. In aeroelastic analysis, aerodynamic modelling methods are divided into panel methods (such as the Doublet Lattice Method (DLM) [37] and the Unsteady Vortex Lattice Method (UVLM) [38]) and higher-accuracy CFD methods. Furthermore, flutter analysis is a key problem in the aeroelastic study of high-aspect-ratio aircraft. Current methods for flutter analysis primarily encompass traditional approaches and their refinements, alongside more novel techniques employing machine learning to predict flutter [39].

Amongst these numerous studies, the majority employ beam models for structural system modelling. This approach necessitates iterative optimisation of model parameters based on experimental results to establish relatively accurate finite element models. For some researchers, lacking experimental conditions makes it difficult to directly and accurately determine beam model parameters, leading to significant deviations in results. Consequently, this method is unsuitable for the extensive precise calculations required during the conceptual design phase. Furthermore, for highly complex systems, the computational cost associated with traditional beam-shell models is often prohibitively high. This paper aims to propose a more efficient and reliable aeroelastic analysis method through the investigation of a complex engineering prototype. Additionally, this paper aims to investigate the effect of structural elasticity on the aerodynamic characteristics and to determine the aeroelastic boundaries of the UAV through the study of an LJS-UAV engineering prototype. On this basis, a structural system analysis model was established using the ‘Simplified beam-shell model’, and the accuracy of the model was verified through mode tests, after which simulation analysis was used to study the aeroelastic characteristics and boundaries of the LJS-UAV.

This paper is organized as follows: Section 2 describes the research subject, discusses the modelling method applicable to this subject, and establishes the aeroelastic analysis model for the whole system. Section 3 conducts a comprehensive simulation analysis of structural characteristics in terms of static aeroelasticity, flutter, and gust response, studies the influence of structural elasticity on the aerodynamic performance of the UAV and the structural dynamic response characteristics under gust disturbances, determines the aeroelastic boundary of the UAV system, and reveals the key structural failure mechanisms. Finally, the discussions and conclusions are summarized in Section 4. The overall flow diagram of the study in this paper is shown in Figure 1.

2. Research Subject Description and Modelling Methodology Study

2.1. Research Subject Description

The geometry of the UAV studied in this paper is shown in Figure 2. The aircraft is divided into the front wing, the rear wing, the fuselage and the vertical stabilizer. The front wing is divided into the inner wing and the outer wing. The inner wing is swept back and inverted. The rear wing is swept front without dihedral. The front and rear wings are connected by the vertical stabilizer and the fuselage.

The global parameters of LJS-UAV are shown in

Table 1. Global parameter.

Global Parameters	Values
Mass	1.5 ton
Cruise Speed	12 m/s
Front Wing Aerofoil	NACA 4412
Rear Wing Aerofoil	Ritz 2-30-14
Front Wing Incidence Angles	1°
Rear Wing Incidence Angles	−2°
Mean Aerodynamic Chord	4 m
Structural Areal Mass Density	2.3 kg/m2

.

2.2. Modelling and Validation of the Structural Finite Element Model

2.2.1. Modelling of the Structural Finite Element Model

In existing research, structural finite element models developed for flexible aircraft primarily include beam-shell models and beam models and solid models. While solid models offer high computational accuracy, their substantial computational demands render them impractical for the aeroelastic analysis of the full-scale aircraft. The Beam models have a relatively low mesh density, but it is difficult to determine the beam parameters rapidly and accurately for a complex system, and often neglect critical structural details. The beam-shell models are widely used in engineering practice. But for highly complex systems, the computational cost of finite element models obtained enjoying detailed beam-shell model (DBSM) can also become prohibitively high, which cannot be accept.

This paper uses the commercial software MSC.Patran 2018 and employs an SBSM for finite element modelling. The SBSM is a system that equates the UAV system to a combination of beams and shells, enabling rapid parameter determination and model establishment through a series of preprocessing tasks. The SBSM method modelling process is similar to the DBSM, but is further simplified based on the DBSM. This method effectively simulates the geometric characteristics and load transfer pathways, with computational costs intermediate between the DBSM and beam model. Given the LJS-UAV’s skin-frame structural design, the following section elaborates on the application of the SBSM in the finite element modelling process.

The primary load-bearing structure of the LJS-UAV wing comprises box beams, with circular tube beams employed at structurally critical regions to augment load capacity. As circular tube beams can be treated as stiffness-equivalent box beams, the UAV’s beam structures are uniformly modelled as box beams. The geometric features of the box girder are extracted according to its actual structural form. The four corners are simulated by beam elements, and the surrounding panels are simulated by shell elements. The material properties for beam elements and shell elements are defined using those of box beams. The cross-sectional shape of beam elements is defined according to the thickness of the box beam’s web plates, while the thickness of shell elements is defined according to the thickness of the box beam’s web plates.

For the rib structures, key points are identified based on beam locations, leading/trailing edges, and supporting stringers. The beam elements are used to simulate the beam after clarifying the geometrical features, and the cross-sectional dimensions of the beams are defined according to the lightening holes and the thickness distributions of the ribs. The Connections between beams and ribs are simulated by beam elements or constraints. The finite element modelling process of beams and ribs is shown in Figure 3. The difference between SBSM and DMSM can also be found in Figure 3. The modelling in this paper was completed using MSC.Patran 2018, employing BAR elements for beam elements and CQUAD4 elements for shell elements. More information about BAR elements and CQUAD4 elements can be found in [40]

The spanwise structures, such as leading/trailing edges and stringers, are simulated by beam elements. The cross-sectional parameters of the beam elements are determined based on the corresponding original structural characteristics. Following this methodology, regions comprising triangular or quadrilateral elements are generated adjacent to the skin, where shell elements simulate the skin. The thickness of the shell element is defined according to the skin thickness. The finite element models of the aerodynamic control surfaces can also be established by this method, with connection stiffness between control surfaces and wing structure determined by design scheme or experimental data and modelled using spring elements. The finite element modelling of the fuselage can be completed by this method similarly.

Some key material properties used in the manufacturing process of UAV are shown in

Table 2. Isotropic material parameter.

Material	$\mathit{E}$ (Mpa)	$\mathit{v}$	$\mathit{G}$ (Mpa)	$\mathit{\rho }$ (kg/m3)
PMI	35	0.3	13	32
Solar Panel	500	0.3	192	350
7050 Aluminium Alloy	70,000	0.3	26,923	2800
Wing Skin	102.6	0.3	39	75

and

Table 3. 2d orthotropic material parameter.

Material	${\mathit{E}}_{\mathbf{11}}$ (Gpa)	${\mathit{E}}_{\mathbf{22}}$ (Gpa)	${\mathit{v}}_{\mathbf{12}}$	${\mathit{G}}_{\mathbf{12}}$ (Gpa)	$\mathit{\rho }$ (kg/m3)
M40J	210	8.81	0.31	4.52	1758
T300	126	8	0.33	3.7	1600

.

Avionics and payload systems onboard the UAV are incorporated into the finite element model as lumped masses or distributed masses. The equipment carried by the drone is shown in

Table 4. The weight of equipment.

Equipment	Weight (kg)
Dynamical System	130
Energy System	550
Flight Control System	100
Payload Equipment	60
Other	40

.

The finalized full-system finite element model is illustrated in Figure 4. Statistically, the finite element model generated by the DBSM method has 322,283 elements and 249,944 nodes, while the model generated by the SBSM method in this paper has 74,096 elements and 38,987 nodes. This represents a 77% reduction in element count and an 84.4% decrease in nodal density compared to the DBSM, significantly improving computational effectiveness.

2.2.2. Mode Test

To validate the accuracy of the finite element model, ground vibration testing (GVT) was conducted on the full-scale UAV system. A hybrid phase resonance-phase separation methodology was employed: the airframe was subjected to random excitation, with resonance frequency distributions determined through phase separation techniques, enabling identification and decoupling of potentially clustered modes. The VXI-640 full-scale airplane GVT system [41] was employed to conduct ground vibration testing. The experimental setup comprised: signal generator, force vector controller for excitation regulation, power amplifier, exciter, 4507B accelerometers (measurable frequency range: 0.3~6000 Hz), Agilent E8403A accelerometer signal conditioners and so on.

In total, 360 sensors were deployed on the aircraft structure to ensure accurate identification of target test modes and enable complete modal shape reconstruction from sensor signals. The installation locations of some sensors are shown in Figure 5. The excitation points were arranged on the front wing, rear wing, fuselage, vertical tail and other components. The placement of exciters is determined by comprehensively considering the location of the nodal line for the modal shape under test, structural symmetry, mass distribution, and actual installation conditions. Adjustments are made as appropriate based on the modal shape being tested to ensure precise excitation of the target modal shape. To simulate the ‘free’ state of the test aircraft, the modal test was carried out using a soft-support suspension system with rubber ropes. The LJS-UAV is suspended horizontally using the aircraft’s five suspension points, with a minimum ground clearance of 0.1 m when hoisted. The suspension method at the junction of the front wing and the rear wing is shown in Figure 6. The highest supported frequency of the airframe is the roll mode frequency, measured in situ as 0.382 Hz. The effect of the suspension system stiffness on the modal frequencies was evaluated and eliminated in the test results listed later.

Following construction and verification of the full-scale test system, comprehensive frequency sweep tests were conducted across the required modal measurement band using multiple excitation force configurations. The resulting spectral curves served as the initial basis for modal identification and separation. Each resonance peak within the spectrum obtained from the sweep testing was identified and separated according to test requirements. Following data storage for the identified modes, the separated modes were subsequently named.

To assess the reliability of the experimentally measured modes, the signals from all measurement points are summarised into a global modal purity indicator function, denoted as $C$. Its mathematical expression is as follows:

$$C=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n\left|R_e\left({\ddot{u}}_i\right)\right|\left|{\ddot{u}}_i\right|}}{{\displaystyle \sum _{i=1}^n{\left|{\ddot{u}}_i\right|}^2}}}.$$

(1)

where:

• $R_e\left({\ddot{u}}_i\right)$ is the real part of the acceleration response at the $i$-th measurement point;
• $\left|{\ddot{u}}_i\right|$ is the magnitude of the acceleration response at the $i$-th measurement point;
• $n$ is the total number of measurement points.

When $C\to 1$, the aircraft can be considered to exhibit a singular vibration mode at the natural frequency, also called a pure mode. A total of 102 orders of modes of the LJS-UAV were cumulatively measured in the test, and all the values of $C$ measured in the test exceeded 0.8, with 66.7% surpassing 0.9, which was a good test result. Approximately 30 global modes with a mode frequency below 10 Hz in the test results.

To evaluate the significance of each mode in structural characteristics, numerical simulation methods were employed to calculate the modal participation factors for the first 30 modes of the UAV across three motion axes: pitch, yaw, and roll. The absolute values of these modal participation factors are presented in Figure 7.

The results from calculating the modal participation factors indicate that the first five modes of the LJS-UAV significantly contribute to its response characteristics under various motion states. Several modes between the sixth and fifteenth orders make minor contributions to the roll motion response characteristics. Subsequent computational analyses in this paper focus primarily on the response characteristics along the UAV’s longitudinal motion. Consequently, the first eight modal orders are selected as the principal modes for comparative analysis and are employed in the aeroelastic calculations presented later.

The first eight structural modal results of the LJS-UAV in the free state (neglect rigid-body motion modes) obtained by using the SBSM, the DBSM and the experimental test are listed, as shown in

Table 5. Structural modal results.

Modal Order	Model Description	SBSM		DBSM		GVT
		$\mathit{f}$ (Hz)	Error (Hz)	$\mathit{f}$ (Hz)	Error (Hz)	$\mathit{f}$ (Hz)
1	Front wing first symmetric bending	0.342	−0.047	0.382	−0.007	0.389
2	Rear wing first anti-symmetric bending	0.614	−0.074	0.671	−0.017	0.688
3	Rear wing first symmetric bending	0.822	−0.161	1.033	0.05	0.983
4	Front wing first anti-symmetric bending	0.877	−0.053	0.968	0.038	0.93
5	Front wing first symmetric torsion	1.171	−0.169	1.339	−0.001	1.34
6	Rear wing second anti-symmetric bending	1.452	−0.086	1.595	0.057	1.538
7	Rear wing bending-torsion coupled mode	1.456	None	None	None	None
8	Front wing first anti-symmetric torsion	1.616	0.034	1.775	0.193	1.582

. The mode shapes are shown in Figure 8.

The seventh-order modal frequency in Table 5 was calculated solely by the SBSM model, with neither the DBSM model nor GVT providing this data: the former yielded no results due to substantial local modal interference during computation, while the latter omitted this mode (encompassing rear wing bending, torsion, and flapping, see Figure 8) from its testing scope, which focused on bending and torsional modes, resulting in data deficiency.

The DBSM demonstrated complete concordance with experimental measurements in modal sequence identification. With the exception of the first anti-symmetric torsional mode of the front wing, which exhibited a frequency error of 12.2%, all remaining modal frequency discrepancies remained below 5%. The modal results obtained from the SBSM were identical to those obtained in the experimental test, with the exception of an order inversion between rear wing first anti-symmetric bending and front wing first symmetric bending. Modal frequency errors obtained using the SBSM are slightly higher than the DBSM. The maximum error between the modal frequency obtained by the SBSM and the measured frequency is only 0.17 Hz.

From a holistic perspective, although the method in this paper does not achieve absolute numerical precision in simulating the LJS-UAV’s modal frequencies, the error of the modal frequency value is low, and the simulation of the modal vibration pattern and the modal order is almost accurate, the relatively low frequency discrepancies (≤12.2% maximum observed error) combined with satisfactory accuracy in mode shape reproduction and modal sequence prediction demonstrate that the developed finite element model effectively captures the global structural dynamic characteristics of the LJS-UAV. This level of fidelity proves sufficient to meet computational assessment requirements during conceptual design phase.

2.3. Modelling and Validation of the Aerodynamic Analysis Models

2.3.1. Modelling of the Aerodynamic Analysis Model

In this paper, the subsonic doublet lattice method (DLM) (for flutter analyses) and subsonic lift surface model (for static aeroelasticity and gust response analyses) provided in MSC.Nastran 2018 software was used to obtain the aerodynamic loads. According to the model geometry and the configuration of the control surfaces, a lifting surface model was established in the MSC.Flds 2018 software. There are 97 elements in the front wing along the span direction, 58 elements in the rear wing, 9 elements in the vertical tail, and 8 elements in the chord direction at all positions. The aerodynamic mesh and the names of all the control surfaces are shown in Figure 9. QZZ1, QZZ2, QZY1, QZY2, HZ1, HZ2, HY1, HY2 are used for longitudinal control, QNZ1, QNZ2, QNY1, QNY2 are drag control surfaces, QWY1, QWY2, QWY3, QWY4, QWZ1, QWZ2, QWZ3, QWZ4 are used for lateral control, and CW1, CW2 are used for heading control.

To solve for the aerodynamic loads, three additional variables must be defined. The first item is the flight dynamic pressure. The section item is the aerodynamic influence coefficient matrix, a function of Mach number and reduced frequency. The third item is the static aerodynamic downwash, which includes the distribution that may arise from an initial angle of attack, camber, or twist.

The flight dynamic pressure and aerodynamic influence coefficient matrix can be defined according to the UAV state. The static aerodynamic downwash can be determined based on the aerofoil camber and the wing twist angle. As shown in Figure 9, the aerodynamic mesh generated within the MSC.Flds 2018 software is effectively composed of planar aerodynamic elements, and defining the static aerodynamic downwash essentially involves specifying the local angle-of-attack for each aerodynamic mesh element. When defining aerofoil characteristics, the mean camber line of the aerofoil is extracted and dispersed according to the distribution of aerodynamic elements along the chord direction. The least squares method is employed to perform linear fitting on each dispersed curve segment, thereby obtaining the slope of each fitted curve segment. This subsequently determines the local angle-of-attack of the original aerofoil at each aerodynamic element. On this basis, by superimposing the initial installation angle of each wing section according to the conceptual design scheme, the static aerodynamic downwash distribution that account for the camber and twist of the aerofoil can be obtained and assigned by the W2GJ card provided by the MSC.Nastran 2018 software, thus adding aerofoil camber and built-in twist to the flat panel model as shown in Figure 10. The W2GJ card will be used in calculations of static aeroelasticity and gust response.

2.3.2. CFD Validation

The effect of aerofoil thickness was usually ignored when defining the static aerodynamic downwash in the absence of experimental methods. To assess the impact of this factor, the aerofoil profiles were extracted from the aircraft. The aerodynamic characteristics of the airfoil were computed using CFD methods, whilst the aerodynamic characteristics of the airfoil with zero thickness were computed using the vortex lattice method (VLM) for comparison. The CFD calculations were performed using the commercial software Fluent 2021R1, with analyses conducted at a flow velocity of 12 m/s and an angle-of-attack increments of 2°. The first layer of the aerofoil mesh was ensured to have a Y+ value of 1. The computations employed the RANS control equations, the S-A turbulence model, and a density-based solver. The aerofoil mesh used for the analysis is shown in Figure 11. The aerofoil mesh comprises a total of 82,260 elements and 82,931 nodes.

The VLM is used to calculate the aerodynamic characteristics of the aerofoil without thickness under the same conditions as the CFD method. The VLM calculations are performed using OpenVSP 3.42.0 software. The aerodynamic characteristics of the airfoil with and without thickness obtained by the CFD method and VLM are shown in Figure 12.

In the data obtained from CFD and VLM calculations, the variation trend of aerodynamic coefficients with respect to angle-of-attack remains largely consistent whether the airfoil includes thickness or not. Within the 0–8-degree angle of attack (AoA) range, the maximum error for the lift coefficient is 12.6% (at 2° AoA), while the maximum error for the pitch moment coefficient is 18% (at 8° AoA). As the VLM calculation process does not consider the effects of fluid viscosity, this results in significant deviations in the drag coefficient. The maximum error occurs at 0° AoA, where the discrepancy exceeds 40%. The structural stiffness of the horizontal swing direction of UAV’s wings is typically highly reliable. During aeroelastic simulation calculations, the focus is usually placed on the bending and torsional deformation of the wings. During structural performance assessment in the conceptual design phase, the lift and moment characteristics of aerofoils influence the overall computational results, whereas drag exerts a lesser impact on the structural response of the UAV. Therefore, a relatively large drag error can be tolerable.

In summary, the aerodynamic data obtained from VLM calculations for the aerofoil is largely consistent with the original aerofoil aerodynamic data derived from CFD computations. Neglecting airfoil thickness does not significantly alter the structural performance assessment outcomes during the conceptual design phase. Consequently, the aerodynamic load analysis model established herein possesses a degree of accuracy, enabling the calculation of aerodynamic performance for LJS-UAVs during the conceptual design stage.

3. Simulation and Analysis of the Aeroelastic Characteristics of the LJS-UAV

The aeroelastic equation in modal coordinates is

$$\overline{M}\ddot{q}+\overline{D}\dot{q}+\overline{K}q=q_{d}Q\left(M_{\infty },k\right)q.$$

(2)

where

$\overline{M}={\mathsf{\Phi }}^{T}M\mathsf{\Phi }$, $\overline{D}={\mathsf{\Phi }}^{T}D\mathsf{\Phi }$, $\overline{K}={\mathsf{\Phi }}^{T}K\mathsf{\Phi }$, $Q\left(M_{\infty },k\right)={\mathsf{\Phi }}^T{G}_{ka}^T{Q}_{kk}\left(M_{\infty },k\right)G_{ka}\mathsf{\Phi }$, $M$ and $K$ are the mass and stiffness matrix of the structure, $D$ is the damping matrix, $Q_{kk}\left(M_{\infty },k\right)$ is the matrix of aerodynamic influence coefficients, $M_{\infty }$ and $k$ are user-defined Mach number and reduced frequency, $q_d$ is the dynamic pressure, $G_{ka}$ is the spline matrix, $\mathsf{\Phi }$ is the modal matrix, $q$ is the modal coordinate vector. The computational model of structure and aerodynamics has been established in the previous section, and the connection between structure and aerodynamics can be established by using the thin plate spline [42] method to solve the aeroelastic equations.

As discussed previously in the structural modal characteristics section, the LJS-UAV exhibits high structural flexibility. When subjected to aerodynamic loads, this inherent flexibility induces substantial geometrical deformation, precipitating a series of aeroelastic phenomena. The unconventional configuration further complicates the structural response characteristics. To facilitate a comprehensive understanding of the LJS-UAV’s aeroelastic characteristics, the systematic investigations were carried out through extensive computational analyses across three critical domains: static aeroelasticity, flutter characteristics, and gust response. This paper employs solutions 144, 145 and 146 provided by MSC.Nastran 2018 [43] to complete these calculations.

3.1. The Static Aeroelasticity Calculation and Analysis

In engineering practice, primary attention is typically devoted to two fundamental categories of static aeroelastic phenomena: the first category encompasses aerodynamic load redistribution and structural torsional divergence; the second category concerns control effectiveness and control reversal phenomena.

For the load redistribution problem, the total aerodynamic lift and pitching moment (head-up is positive) of the entire aircraft under wind tunnel (with fixed center of gravity) across varying flow states were calculated for the velocity range of 0–40 m/s and the AoA range of 0–6°. Meanwhile, the aerodynamic data of the rigid aircraft configuration were calculated by the CFD method for comparison, and the results are shown in Figure 13.

The aerodynamic forces and moments of a rigid aircraft are proportional to the square of the incoming flow velocity and linearly related to the angle of attack within small incidence ranges. The aerodynamic characteristics of a flexible aircraft change considerably due to the effects of structural elasticity.

For the flexible configuration, the lift is positively correlated with the angle of attack, but the slope of the lift curve appears to increase and then decrease as the incoming flow velocity increases. The pitching moment increases with the rise in the incoming flow velocity, the positive moment first increases and then decreases to become a negative moment, after which the absolute value of the negative moment first increases and then decreases to become a positive moment, and finally keeps the trend of the positive moment gradually increasing until the structural divergence.

To investigate the mechanism of this phenomenon, structural deformation data under varying flow states were extracted and analysed. Since the deformation of the wing structure in the bending direction has a little effect on the aerodynamic forces and moments, this paper focused on torsional angle variations to elucidate the causal relationship between structural configuration and aerodynamic response. The torsional deformation characteristics of the front and rear wings are shown in Figure 14.

Apart from localised regions of the rear wing, the entire wing structure exhibited consistent torsional deformation patterns. The rear wing’s outer sections demonstrated minor incidence angle fluctuations within a ±4° range, exerting negligible influence on the overall lift. Conversely, the torsion angle of the front wing and the inner wing of the rear wing change in a basically consistent trend, which determines the overall lift to a large extent. In low speeds (<7 m/s), rapid torsional growth in the flexible aircraft produced a steeper lift-dynamic pressure gradient than the rigid configuration, exceeding rigid-aircraft lift predictions. In medium speeds (7–30 m/s), torsional growth attenuated progressively. Lift became governed by dynamic pressure and torsion, with torsional effects remaining dominant, causing a gradual smaller rate of increase in lift. Above 30 m/s, the dynamic pressure dominated, and the effects of torsion were deduced. Lift was nearly proportional to speed squared until the structural divergence. The interpretation of the variation in the moment is similar to that of the lift force.

The longitudinal aerodynamic derivatives were calculated, and the results are shown in Figure 15.

For rigid aircraft configurations, aerodynamic derivatives remain essentially constant with increasing velocity. In contrast, the flexible aircraft exhibit significant wing bending and torsional deformation under rising dynamic pressure, leading to calculated aerodynamic derivative variations. At design dynamic pressure conditions, the flexible configuration’s lift curve slope $C_{L_{\alpha }}$ attenuates to 47.8% of the baseline rigid geometry value.

As the flight velocity increases, the effectiveness of various control surfaces decreases to varying degrees. The LJS-UAV features aerodynamic control surfaces on both front and rear wings, enriching the strategy of longitudinal control. As each control surface experiences differential flow interactions depending on its location, this paper quantitatively evaluates control derivatives across varying flow velocities. Control effectiveness (denoted ‘$\eta$’) is defined as the ratio of the flexible aircraft’s control derivatives to those of the rigid configuration. Its mathematical expression is as follows

$$\eta ={\displaystyle \frac{C_{m\delta e-flexible}}{C_{m\delta e-rigid}}}.$$

(3)

where $C_{m\delta e-flexible}$ is the longitudinal control derivative of flexible configuration, and $C_{m\delta e-rigid}$ is the longitudinal control derivative of rigid configuration. The results are shown in Figure 16. Due to the symmetry of the aircraft, only results from control surfaces on a unilateral wing are illustrated.

The $\eta$ diminishes rapidly with increasing velocity. At the design cruise speed, the $\eta$ of QZY1 and QZY2 decayed to 52.4%, while the $\eta$ of HY1 and HY2 decayed to 29.5% and 40.4%. When the speed exceeds 18 m/s, the rear wing control surfaces sequentially demonstrate control reversal phenomena, whereas the front wing control surfaces show no such reversal within the calculated speed range.

When structural elasticity is disregarded, the rear wing control surfaces exhibit higher $\eta$ than the front wing due to the longer distance from the center of gravity. However, when considering the effect of structural elasticity, the $\eta$ of the rear wing control surfaces degrades rapidly with increasing velocity. By checking the control derivatives of the front and rear wings’ control surfaces (HY1 and QZY1) in the flexible aircraft configuration at the design cruise speed, it is found that they are basically the same. Nevertheless, the front wing control surfaces maintain higher effectiveness and demonstrate greater resistance to control reversal under further increases in freestream speed.

To further assess the longitudinal trim characteristics of the front/rear wing’s control surfaces configuration, static aeroelastic trim calculations were conducted at varying flight velocities under free conditions with a load state of 1 g. Structural constraints were imposed on the fuselage structure to limit the aircraft’s motion in yaw and roll directions. The rigid-body reference frame adopted the body-axis system, with computations performed at sea level altitude. The AoA and control surface deflections were solved by fixing the front or rear control surfaces, respectively. The control surfaces to be solved are deflected in the same direction and at the same angle. The calculated structural divergence speeds and corresponding modes are shown in Figure 17. When the control surfaces of the front wing or rear wing were used alone for trim, the aircraft’s divergence is shown as the torsional divergence of the front wing or the bending divergence of the rear wing. The front wing’s superior stiffness characteristics yield a divergence speed through trim analysis that demonstrates a 90% enhancement compared to that of the rear wing.

From the preceding discussion, it is evident that the rear wing control surfaces exhibit control reversal phenomena at a comparatively lower speed, concomitantly reducing the aircraft’s static aeroelastic divergence speed. Consequently, the operational reliance on the rear wing control surfaces for flight control should be reduced. However, the practical implementation of sole reliance on front wing control surfaces for trim operations necessitates either enlarged control surface dimensions or excessive deflection angles. Therefore, the strategy of active control of the front wing control surfaces and auxiliary control of the rear wing control surfaces should be adopted to complete the longitudinal stabilisation and control of the aircraft in practical applications.

3.2. The Flutter Characteristics Calculation and Analysis

Chen [44] pointed out that the interaction and constraints between the front and rear wings of joined-wing aircraft cause their flutter characteristics to be characterised by multi-modal participation. The p-k [42] method for flutter analysis is used in this paper to accurately predict the flutter boundaries using an unsteady aerodynamic model based on harmonic motion. The reduced frequency range is determined by the first 8 orders mode frequency of the aircraft. The range of modal frequency is 0~1.6 Hz. The range of reduced frequency at sea level is 0~1.9. The mode shapes are shown in Figure 8, and the mode frequencies are shown in Table 5.

Frequency-domain flutter characteristics were calculated for the aircraft at altitudes of 0, 5, 10, 15, and 20 km. The air density corresponding to each actual flight altitude was employed in the computations. The v-g and v-f curves for the calculated cases are presented in Figure 18. It is generally believed that when the g value of a mode in the v-g curve exceeds 0, structural flutter occurs. The corresponding speed is the flutter speed. The frequency of this mode in the corresponding v-f curve at this speed is the flutter frequency. Computational results indicate that at sea level, the aircraft’s matched flutter speed [42] is 27.5 m/s. Flutter occurs in the front-wing symmetric first torsional mode, with a flutter frequency of 0.585 Hz.

Flutter phenomena in UAVs typically result from coupling between two structural modes. However, sea-level computational analyses revealed only a gradually close frequency convergence between the front-wing symmetric first torsional mode (Mode 5) and the rear-wing bending-torsion coupled mode (Mode 7). The rear-wing bending-torsion coupled mode is asymmetric and does not occur during flight. This phenomenon will be discussed further in Section 3.3.2.

Computational results at 10 km and 20 km altitudes distinctly demonstrate that coupling between the rear-wing first symmetric bending (Mode 3) and the front-wing first symmetric torsion (Mode 5) induces divergence in the latter torsional mode, ultimately triggering flutter. The results at 5 km and 15 km are similar to the results at 10 km and 20 km.

As flight altitude increases, the flutter modes transition from Modes 5 and 7 to Modes 3 and 5. It can be determined that Mode 5 consistently participates in the flutter at all flight altitudes. Observing the frequency trend of Mode 5 in Figure 18 reveals that as altitude increases, the variation in Mode 5 frequency with increasing flight speed remains relatively stable. At sea-level altitude, increasing flight speed causes Mode 3’s frequency to decrease rapidly without coupling to Mode 5, while Mode 7’s frequency also decreases rapidly but couples with Mode 5. At 20 km altitude, increasing flight speed causes the frequencies of Modes 3 and 7 to change from rapid decrease to gradual increase. Consequently, Mode 3 couples with Mode 5 to initiate flutter, whereas Mode 7 does not couple with Mode 5. This indicated that both Mode 3 and Mode 7 could couple with Mode 5 to initiate flutter. However, at different flight altitudes, the influence of flight speed on Modes 3 and 7 differed, leading to a transition in the flutter modes from Mode 3 coupled with Mode 5 to Mode 5 coupled with Mode 7.

To investigate the influence of flight altitude on the UAV’s flutter characteristics, the results at different flight altitudes are shown in

Table 6. The flutter results at different flight altitudes.

Altitude (km)	Matched Flutter Speed (m/s)	Flutter Frequency (Hz)	Flutter Dynamic Pressure (kg·m−1·s−2)
0	27.5	0.586	463.2
5	30.6	0.661	344.8
10	35.2	0.717	256.2
15	45.2	0.740	198.9
20	61.0	0.723	165.4

.

The matched flutter speed exhibits progressive elevation with increasing flight altitude, accompanied by an accelerating rate of increase, but the flutter dynamic pressure demonstrates sustained attenuation. The design cruise speed of the aircraft at sea level is 12 m/s, which reaches 44.5 m/s at 20 km altitude through equivalent dynamic pressure calculations, and is below the computed flutter speed of 61 m/s at 20 km altitude. Although the calculated flutter dynamic pressure decreases with the increase in flight altitude, the design cruise state of the flight has been kept within the flutter boundary.

With the increase in altitude, the flutter frequency increases from 0.585 Hz to 0.74 Hz and then has a decreasing tendency. The flutter frequency obtained at different altitudes does not exceed 1 Hz, which indicates that the aircraft’s structure is extremely flexible and very sensitive to external disturbances. But the cruise flight state is within the flutter boundaries, so the structure can be considered reliable.

3.3. The Gust Response Calculation and Analysis

The LJS-UAV is characterised by large structural flexibility and geometric scales, which make it highly susceptible to gust loads and can even lead to structural failures. The 1-cos gust model is widely used in the study of structural dynamics, the design of flight control systems and the evaluation of flight quality. The 1-cos gust model [45] is used in this paper, as shown in Figure 19.

3.3.1. The Analysis of Gust Response Characteristics

Solar-powered UAVs subjected to gust loading exhibit substantial structural deformation, inducing alterations in aerodynamic characteristics that compromise control system efficacy. In order to evaluate the sensitivity of the UAV to gusts, the gust response of the LJS-UAV at sea level was calculated using the 1-cos gust model.

The gust frequencies are 1–8 Hz, and the gust velocities are 3, 5, 7, and 10 m/s. The wingtip leading edge node was taken as the observation node, and the maximum magnitude of vibration after loading of the node under different computational conditions is extracted as shown in Figure 20.

From an energy perspective, increasing gust frequency reduces the time of structural excitation, and the gust energy decreases. When the gust velocity amplitude increases, the time acting on the structure is the same, but the amplitude is higher, which leads to a rise in the gust energy. Therefore, the vibration amplitude decreases with the increase in gust frequency and increases with the rise in gust velocity. When the gust frequency is 1 Hz, the maximum amplitude of the vertical direction of the structure caused by the 3 m/s gust is 2.5936 m (about 7.8% of the half wingspan), which causes the dihedral of the wing to increase by about 4.45°. When the gust velocity increases to 10 m/s, the maximum amplitude of the vertical direction of the wingtip reaches 8.6486 m (about 26% of the half wingspan), and the dihedral of the wing increases by 14.56°. The shape of the aircraft changed considerably.

Four sections at the wingtip, the front and rear wing junction, and 50% of the span of the front and rear wings are taken as monitoring surfaces, named Section 1, Section 2, Section 3, and Section 4. The position of the sections is shown in Figure 21.

When the gust velocity is 5 m/s and the gust frequency is 3 Hz, the AoA and displacements of the monitoring surface are shown in Figure 22.

The deformation response from wing tip to wing root will be uniform in a conventional configuration aircraft. In contrast, results for the LJS-UAV reveal distinct characteristics. The AoA of Section 1 oscillated within 5° after being affected by the gust. The range of section AoA oscillations decreases significantly as the monitoring surface approaches the aircraft’s symmetry plane. The AoA of Section 2 oscillated within 1°, and the maximum value of the AoA oscillation is 87% smaller than Section 1. The maximum displacement is 49.4% smaller than Section 1. This comparative analysis demonstrates that the structural deformation of LJS-UAV after being affected by gusts mainly occurs in the outer wing section. The mutual structural restraint between front and rear wings effectively suppresses the torsional deformation of the inner wing, reducing the effect of gusts on the structure.

3.3.2. The Analysis of Time Domain Flutter Characteristics

Conventional flutter analyses are performed in the frequency domain using the doublet lattice method to compute unsteady aerodynamic forces, which typically delivers high predictive accuracy. However, in the results of Section 3.2, an anomaly was found that the structural flutter modes changed due to the modal frequency decreasing to 0 Hz. In this section, the structural generalized coordinate displacement response in the time domain was calculated to determine if flutter occurs. The results obtained using this method are more intuitive.

When the airspeed of a UAV is below the flutter speed, the structural system is a stable system. The structural response will gradually converge due to the presence of damping after being disturbed. When the airspeed increases to a certain velocity, the elastic structure undergoes self-excited vibration with non-decaying amplitude due to the coupled effects of aerodynamic forces, elastic forces, and inertial forces. At this point, any external disturbance will cause the elastic structure to vibrate with the same amplitude, indicating that the structure has entered flutter. The time-domain response behaviour of the structure at this point represents the mechanism by which the elastic structure undergoes flutter. This paper applies disturbances to elastic structures in the form of vertical gusts, calculates the time-domain response of the elastic structures after the disturbances, finds the flight speed at which the elastic structures undergo vibrations of equal amplitude, and determines the flutter boundary of the UAV.

The computational analysis was conducted at sea level, with a gust velocity of 3 m/s and a gust frequency of 1 Hz. The wingtip leading-edge node was selected as the monitoring point. A structural flutter boundary search was performed across a flight velocity range of 10–40 m/s, identifying three representative flight velocity states with distinct dynamic responses: 26 m/s, 28.13 m/s, and 28.5 m/s.

The vertical displacement response and acceleration response curves of the monitoring point are shown in Figure 23.

When the flight velocity is below 28.13 m/s, the wingtip oscillations gradually attenuate following a disturbance. At the critical velocity of 28.13 m/s, sustained constant amplitude oscillations in the vertical displacement of the wingtip are observed. Structural response analyses under gust excitations of varying amplitudes and frequencies at this velocity state reveal that any gust input induces undamped oscillations, indicating that the system has reached its flutter boundary. The time-domain method identifies a flutter velocity of 28.13 m/s and a flutter frequency of 0.36 Hz, consistent with frequency-domain flutter analysis results. For the flight velocity exceeding 28.13 m/s, any disturbance induces rapidly diverging wing displacements and accelerations.

The flutter is typically initiated by the coupled oscillatory divergence of bending and torsional deformations. To investigate the root cause of structural vibration divergence in the LJS-UAV, displacement contour plots at discrete time intervals during constant-amplitude oscillations were generated, as shown in Figure 24.

The results show that there are bending, torsional and waving motions in the front and rear wings during the oscillation. During forward waving motion, the wing experiences downward bending and negative torsional deformation, reducing the effective AoA and consequently reducing aerodynamic loads. The elastic restoring forces arising from horizontal waving and vertical bending drive the structure toward its equilibrium position. An upward inertial force is generated when crossing the equilibrium position. The wing experiences upward bending with backward waving and positive torsional motion, increasing the AoA and amplifying aerodynamic loads. Repeating this process over and over creates the oscillations. Thus, for the LJS-UAV, flutter is not exclusively driven by bending and torsional motion but rather emerges from the coupled effect of bending, torsional, and waving motions.

The displacement response curves of the wing tip monitoring point in the XZ plane for different incoming flow velocities are shown in Figure 25.

Following gust excitation, the aircraft exhibits substantial horizontal and vertical displacements across all states, but there is a significant difference in the structural response after the gust excitation stops. In the converging state, both horizontal and vertical displacements decrease and converge quickly, eventually fluctuating within a small range. In the constant amplitude oscillation state, both horizontal and vertical displacements are oscillating at a certain amplitude, and the structural response of the monitoring point shows a form of elliptical motion that continues. In the divergent state, the structure returns to its initial position soon after being disturbed by a gust, followed by a self-excited oscillation, with both horizontal and vertical displacement increasing and the response curve acting like a progressively enlarged ellipse. The results in the ZX plane reveal that structural vibration divergence is driven not merely by the vertical bending and torsional motions, but also by the horizontal waving motions, which should not be neglected.

4. Conclusions

This paper focuses on the LJS-UAV engineering prototype. A computationally efficient aeroelastic analysis model was developed using an SBSM, whose accuracy was validated through a structural mode test and CFD simulations. The LJS-UAV was confirmed to have extremely rich aeroelastic characteristics by the mode test and comprehensive simulation analysis. This research investigates the effect of exceptionally flexible structures on aerodynamic performance, along with their response behaviour under gust disturbances. Finally, the aeroelastic boundaries of the LJS-UAV were conclusively established. The main conclusions are as follows:

• The adoption of an SBSM for establishing an aeroelastic analysis model enables sufficient simulation of structural performance while simultaneously reducing the number of finite element model elements by 77% and nodes by 84.4%, compared to the DBSM, significantly enhancing computational efficiency. The mode test confirmed that the SBSM is suited for the assessment of structural performance during the conceptual design phase of the LJS-UAV.
• The LJS-UAV exhibits 30 global modes with frequencies below 10 Hz, indicating extreme structural flexibility. As flight speed increases, torsional deformation at the wing root exerts a decisive influence on the variation trends of lift and moment. Structural deformation during flight induces pronounced variations in aerodynamic derivatives. Consequently, the effects of structural elasticity must be given due consideration in the design of the control system.
• Owing to structural elasticity, both the control allocation strategy and control surface efficiency significantly influence the aeroelastic boundaries of the LJS-UAV. The adoption of control surfaces, mounted on the front wing, which has a greater stiffness, for longitudinal trim and control, is conducive to improving structural performance and expanding the flight envelope.
• Due to experimental constraints, the study in this paper was limited to validating the model’s accuracy through a mode test. It is expected that researchers will have the opportunity to use more advanced experimental methods for validation, or to consider the structural geometric nonlinearities and rigid-body degrees of freedom motion for more in-depth studies.