In aerospace structural design, the vibration properties of aircraft wings are still a major concern due to their direct effect on aeroelastic stability, fatigue life, and hence the safety of flight. Interference among the structural stiffness, aerodynamic forces as well as inertial effects yields a complex bending and torsion coupling behavior which could result in resonance and flutter if not properly eliminated (Patil and Patil, 1997). The precise estimation of modal parameters, which includes natural frequency, mode coupling and deflection shapes, is therefore central for ensuring structural reliance based on variable condition of flight. In the last three decades, an understanding of the dynamic response of wings with varying platform geometry, martial composition and internal structures has significantly progressed. The analysis conducted earlier by (Patil and Patil, 1997; Jha et al., 1997) established the basics of aeroelastic coupling in composite wings, the matter which reveals how the angle of sweep influences the critical flutter speed. A continuous development, specifically the introduction of the FEM and the dynamic stiffness method (Pagani et al., 2014; Banerjee, 2016; Viglietti et al., 2017), enable the highly precise prediction of modal behavior in slender and multi-cell wing structures.

The spar element plays the key role in reducing wing vibration. Previous studies (Sedaghati et al., 2006; Miskin and Takahashi, 2019; Pany et al., 2001) revealed that box spars provide superior torsional stiffness and higher natural frequencies than their counterparts I- shaped spars, which have in turn lower deflection values. The analytical and experimental studies validate such behavior (Demirtaş and Bayraktar, 2019; Hoy et al., 2023; Pany, 2023). The effect of material anisotropy and structural arrangements on the fluttering margin was documented to reduce it by about 20% (Guo et al., 2006; Jonsson et al., 2023; He et al., 2023).

Nonlinear behavior of tapered swept wings was investigated by (Patuelli et al., 2024; Elshazly et al., 2025) using computational fluid dynamics- coupled aeroelasticity. Studies on biomimetic configurations (Basak and Akdemir, 2024) and optimizing composite wings (Rajamurugu et al., 2024) showed that innovative selection of geometric characters leads to simultaneous improvement in aerodynamic performance and dissipating vibration energy. Despite this progress, a comprehensive comparison addressing the combined influence of wing geometry (tapered vs. swept) and spar configuration (box vs. I-section) under uniform boundary and material conditions remains limited.

To fulfill the gap in the existing literature, the current work investigates the influence of different geometries of wings on vibration behavior using FEM. Two types of symmetric wings (taper and swept) supported with box and I-section spars using 6 NACA (The National Advisory Committee for Aeronautics) wing types (0024, 2411, 2416, 2424, 4412, 4421) were implemented. Furthermore, natural frequencies, mode shapes and deflection behavior under a certain structure and aerodynamic parameters were evaluated. The main objective was to quantify how the geometric and internal design parameters synergistically shape the model behavior of aircraft wings, providing a clear understanding of their potential for aeroelastic optimization. The models studied were carefully selected to highlight the two main parameters: wing thickness and curvature in terms of camber values. The addressed cases had a variable thickness-to-cord ratio that ranged from 11% to 24% while the deviation in camber was 4%. A concise summary of key contributions from previous studies is presented in Table 1, highlighting methodological evolution, comparative findings, and persistent research gaps that underpin the motivation for the present analysis.

This section outlines the numerical methodology employed to investigate the influence of wing geometry on vibration characteristics through a comprehensive FEM analysis. The simulations were performed using ANSYS Parametric Design Language (APDL R15), which provides a reliable platform for extracting natural frequencies and mode shapes in complex three-dimensional aerospace structures. This approach allows accurate assessment of the structural stiffness and modal coupling effects associated with different wing configurations. The aerodynamic behavior of each wing was analyzed using the thin airfoil theory and Prandtl’s lifting-line theory, which relate the aerodynamic circulation to the angle of attack and spanwise lift distribution for finite wings.

The input data are aircraft weight, air density, speed and NACA wing models, their planform, cord length and angle of attack to evaluate the aerodynamic coefficients for each configuration based on these theories to characterize the adequate wing length. These values are considered the light weight - general aviation and training airplanes, where their working characteristics are: taper ratio (0.35–0.6), speed (40–75 m/s) in addition to the weight range (600–1,200 kg) and cord length (0.9–1.2 m).

The aerodynamic coefficients were determined for each configuration based on these theories to characterize the aerodynamic loading conditions applied within the FEM environment. The structural and aerodynamic analyses were integrated within a unified computational framework to find the combined influence of geometry, material properties, and boundary conditions on the modal response. Two primary wing planform configurations, tapered and swept-back, were modeled under identical mechanical and aerodynamic parameters to isolate geometric effects. Modal parameters, including natural frequencies, mode shapes, and deflection patterns, were extracted for each configuration. All numerical results were validated against established analytical data, confirming the accuracy, convergence, and physical reliability of the developed model.

The governing equations the motion of the fluid domain are the incompressible Navier-Stokes equations, and the equations that govern the structural response are the dynamic equilibrium equation of elasticity. The fluid equations (Equations 1, 2) can be formulated as: ∇ · u = 0 (1) ρ f f ∂ u / ∂ t + u u − u m · ∇ = − ∇ p + μ f ∇ 2 u (2)

Where u is the fluid velocity, um is the mesh velocity (from structural deformation), p is pressure, and ρ_f, μ _f are the fluid density and viscosity.

The structural dynamics of the deformable airfoil surface are governed by the equation of motion (Equation 3) for elastic solids: f e x t = C ∂ d / ∂ t + ρ s ∂ 2 d / ∂ t 2 + K d (3)

Where d is the displacement vector, ρ_s is the structural density, C is the damping matrix, K is the stiffness matrix and f_ext represents the external aerodynamic loads transferred from the fluid.

As air passes around the wing, and due to its angle of attack and camber, with the presence of dynamic pressure from speed and density, a pressure difference occurs between the upper and lower wing surfaces. The resultant force directed between the wind direction and the vertical, acting as lift, is translated by the wing area into a two-component force: lift and drag. This is the fundamental principle of flight. Swept-back wings are better in comparison to other wings, especially to swept-forward wings in their drag force. Laboratory and on-site tests used to compare wing performance face the problem of the variations in size and shape between different wing types. Therefore, it was necessary to find a standard that reduces differences and unifies the method of evaluating wings in terms of lift, drag, and torque. This is why the lift and drag performance of wings is evaluated using dimensionless numbers that minimize the effects of shape differences (Ahmed et al., 2025). Wing coefficients are dimensionless dynamic coefficients derived using the Fast Fourier Transform (FFT) to relate the angle of attack to the rotation around the wing, based on the ideal potential flow state of constant density, with no internal friction and no rotation (Liu, 2021). The general form of the lift coefficients is presented in Equations 4–7 as below. C L = a α − α 0 (4) a = a 0 1 + a 0 π   A R   e (5) C D = C D 0 + k   C L 2 (6) k =   1 π   e   A R (7)

Where C_L is the coefficient of lift, C_D is the coefficient of drag and α is the angle of attack, C_D0 is the coefficient of drag at ( α 0 zero lift angle of attack, while the slop of the curve of coefficient C_L versus angle of attack, AR is the aspect ratio and e is Oswald constant, a 0 Two-Dimensional Airfoil Lift Slope

The three main structural components of the wing, spar, ribs, and skin, provide an ideal balance between strength, flexibility, and load distribution. Spar, with its high resistance to bending and torsion, is the reason for the wing’s stability and resistance to dynamic forces. Ribs transfer forces from the wing surface to the spar and support the skin surface shape, providing high flexibility and wing surface dent resistance. Skin distributes aerodynamic forces smoothly and contributes to enhanced torsional rigidity. The synergy of these three parts gives the structure high flexibility, uniform force distribution, and reinforcement, while also achieving efficient mass distribution.

In general, aircraft are designed to meet specific requirements and need efficient aerodynamic conditions to achieve them, along with exceptional structural resilience to survive and withstand these conditions. Accordingly, the design process determines the wing surface shape, cross-section, position on the fuselage, length, root width, and taper. The aircraft’s maneuverability, speed limits, stability, payload, runway length, takeoff and landing performance, are additional constraints that place strict limitations on the internal structure and external shape of the wings (Jaafar and Hmoad, 2024).

Wings are classified depending on wing surface shape or what so called the planform front view, position on the fuselage, and their cross sections. The main types of wings are rectangle, taper, swept back, swept forward, dihedral, anhedral, and they may be low mounted position, or highly mounted wings, they may be elliptical, and it is a long list to be counted. There are some points that are related to wing aspects among them, all wings consist of spars, ribs, and skins, and all increase life forces by increasing their surfaces, which in turn lower the ability for maneuverability. In this work, two types of wing planforms will be studied, these are: tapered and swept-back wings, having different NACA models and two types of spars under the same aerodynamic and structural conditions to isolate the influence of planform geometry on natural frequencies, deflection patterns, and mode-shape coupling characteristics.

In this study, two principal wing geometries were analyzed under the Aircraft overall weight is 1,100 kg, the speed is 50 m/s and ρ = 1.225 kg/m³ to isolate the effect of planform shape on overall span length. The configurations considered were:

Tapered wing

Each configuration was modeled for multiple NACA airfoil sections and evaluated at and spar section. The aerodynamic and geometric parameters were determined to ensure equivalent lift generation and structural loading across all cases (Mostakim et al., 2020).

The three-dimensional wing geometries were generated in ANSYS Mechanical APDL (Release 15) using the aerodynamic parameters summarized in Tables 2, 3 for six selected four-digit NACA airfoil profiles. Each of the four digits has its own meaning. For example, NACA-2411 has a maximum distance between the cord line and the mean line of 2% of the cord length, and so-called camber or curvature localized at 4% of the cord length with a thickness to cord length ratio 11%. The airfoil section was defined by its x – y coordinate data, obtained from publicly available NACA databases in CSV format. The coordinates were imported into ANSYS and converted to key points, which were subsequently connected using spline curves to reconstruct the exact airfoil contour, as illustrated in Figure 1.

The resulting two-dimensional profiles were extruded along the spanwise direction to generate the wing skin with a uniform thickness of 1.6 mm and a taper ratio of 0.4. To simulate the internal structural framework, ribs must be spaced according to some reliable standard, and it is documented that rib spacing could achieve a good balance between weight and wing strength if it is around 25%–50% of the wing cord length (Arunkumar et al., 2012). Eleven equally spaced ribs were introduced for both tapered and swept configurations. Each wing was reinforced by either a box spar or an I-section spar, whose geometric properties are listed in Tables 4, 5, respectively.

ANSYS Parametric Design Language (APDL) script was developed to automate the construction of 3-D wing models. The script sequentially defined airfoil coordinates, spline surfaces, ribs, and spars, and then extruded and mirrored the geometry to form the full wing structure. Table 6 lists the sequential input and processing steps to build up 3-D NACA wing geometry.

The use of step-by-step programming in ANSYS numerical analyzer environment offers several advantages: all models are represented in a consistent manner, eliminating variations due to approximation or human error, and ensuring faster execution. Sample results from using the program are shown in Figures 2–4.

Among the most well-known experimental research techniques for measuring wing performance and flexibility is structural or Ground Vibration Testing (GVT), which addresses structural aspects, and wind tunnel tests to evaluate aerodynamic performance. The accuracy of these methods is subject to several limitations and challenges, including high cost, the need for time recording and preparation, and, most importantly, the requirement to take the aircraft out of service for the research, in addition to numerous human factors. The FEM overcomes these problems with an acceptable approach, a small margin of error, and the ability to study a wide range of engineering aspects (Mousa et al., 2022). In the present study, FEM was used to determine the natural frequencies and mode shapes of various wing geometries in order to assess the influence of planform and spar configuration on vibration characteristics.

The most in use material in aircraft manufacturing sectors is aluminum alloys due to the superior strength to weight ratio. Aluminum mechanical properties and its behavior are subjected to different tests in order to be carefully fit analytically. Results and previous works documented that aluminum use in airplane industry and their simulations are matched under linear elastic material assumptions.

The behavior of aluminum alloys used in aircraft vibration modeling is treated as linearly elastic, isotropic, and homogeneous. The rationale behind these assumptions is that the aircraft is designed to operate without plastic deformation, which is itself a cause of wing failure, and that its properties are sufficiently uniform to be considered practically isotropic and homogeneous. Experiments have shown that the margin of error resulting from these assumptions is small (between 1% and 5%), and they can be relied upon, especially for research purposes, without the need for complex nonlinear models (Doğan and Şahin, 2021).

Each wing model consists of three major structural components, skin, ribs, and spars, constructed from aluminum alloys commonly used in aircraft structures. The material properties of these components are listed in Table 7. All materials were modeled as linear elastic and isotropic, an assumption that is valid for the small-deformation regime considered in this analysis (He et al., 2023).

All models were discretized using SOLID186, a 20-node hexahedral (brick) element capable of modeling irregular geometries and supporting both structural and coupled-field analyses. SOLID186 elements are well-suited for modeling aerospace structures, gear drives, metallic frameworks, and thermal–structural problems due to their ability to handle both linear and nonlinear material behavior as well as contact interactions. Each node in this element has three translational degrees of freedom (in the x, y, and z directions), resulting in a total of 60 degrees of freedom per element. In general, the rounding and discretization errors are the two main errors encountered with the choice of element type so that it must be obeyed a certain procedure to avoid such errors. Many elements are adequate to fit the problem and reach the solution but with different accuracies like quad, brick and tetrahedral elements. Here, brick elements are adopted to simulate the natural frequency. The right element number will be chosen according to the convergence test to achieve good balance between the discretization and round-off errors by increasing the number of elements gradually till it reaches 14,000 elements; to confirm the results of previous work as it will be discussed in verification subject. The geometry of the brick element is illustrated, and the fully discretized 3-D finite element model, in addition to the FEM solution flowchart, is shown in Figure 5.

The current research has studied how the wing geometry and structural configuration can affect the vibrational characteristics of aircraft wings, using the FEM. The six NACA profiles of airfoils were examined under two major planforms, tapered and swept-back, with box-spar and I-section spars. Modal and deflection analysis was conducted in order to find the natural frequencies and deformation. Further, two-way ANOVA was also done in order to verify the level of statistical significance of the differences in the results achieved. The principal conclusions obtained based on the findings are:

The tapered wings had greater natural frequencies than the swept-back wings in the six NACA investigated profiles. This is primarily because of the reduction in the effective spans and increased chordwise stiffness. The mean first mode frequency distribution of the tapered setup (9.75 Hz) was nearly 25% higher than the swept wings (7.3 Hz).

Based on the obtained findings, it can be concluded that the tapered wings supported with I-section spars are the most favorable with weight efficiency, stiffness and vibration stability. In the other hand, swept box-spar configurations offer more flexibility beneficial for the high-speed applications. Since, the present work is limited to the linear elastic analysis; the nonlinear geometric and aeroelastic coupling effect might further affect the dynamic response. Also, further studies have to take into account the integrated Fluid-Structure Interaction (FSI) simulations as well as experimental modal testing in order to verify and extend these results under realistic flight conditions.