Document Type : Research Paper
Authors
^{1} Faculty of Mechanical Engineering, University of Kashan, Kashan, 8731753153, Iran
^{2} Faculty of New Sciences and Technologies, University of Tehran, Tehran, Iran
^{3} Faculty of Engineering, Yazd University, Yazd, Iran
Abstract
Keywords
Main Subjects
Research Article
Ahmad Reza Ghasemi ^{a*} , Hamid RabieyanNajafabadi ^{b},
Hossein Nejatbakhsh ^{a}, Amin Gharaei ^{c}
^{a} Faculty of Mechanical Engineering, University of Kashan, Kashan, 8731753153, Iran
^{b} Faculty of New Sciences and Technologies, University of Tehran, Tehran, Iran
^{c} Faculty of Engineering, Yazd University, Yazd, Iran
ARTICLE INFO 

ABSTRACT 
Article history: Received: 20230519 Revised: 20231205 Accepted: 20240112 

The aeroelastic stability of the tail is significantly challenged by flutter instability. Skin and spars strongly affect flutter speed due to their torsional and bending stiffness, respectively. Csection spars are primarily utilized in composite structures due to their straightforward manufacturing process. This research aims to investigate the impact of the position and orientation of a laminated composite Cspar on the flutter speed of the airfoil section, utilizing a twodegreeoffreedom flutter method. The position of the Cspar varies between 10% and 50% of the chord length from the leading edge of the airfoil section, while the orientation of the Cspar with respect to the leading edge or trailing edge is also examined. To ensure comparability, the elastic section modulus and mass of the composite spar are maintained nearly constant. When it comes to the structural design process, one of the key challenges is determining the flutter and divergence speeds. In a novel approach, Finite Element Method (FEM) is utilized to calculate the torsional and bending stiffness values. This method provides a more accurate and efficient way to evaluate these important parameters. The results indicate that the location design of the Cspar exerts a more substantial influence on the flutter speed than the orientation of the spar. Furthermore, it is crucial to consider the nonlinear effects of the spar's position and direction in comprehending the aeroelastic instability of the aircraft tail. Additionally, the study found that the addition of a spar to a hollow section of the Vtail does not significantly enhance aeroelastic behavior. Only a modest increase of approximately 20% in flutter speed was observed. The primary effect of the spar lies in the bending stiffness, which does not lead to a substantial increase in flutter speed. Moreover, while flutter occurs before divergence, there can be a considerable distance between the respective speeds. Moving the spar from the leading edge to the midchord can reduce this margin, potentially compromising stability. Results show when the Cpar position is close to the center of the airfoil, the flutter and divergence speed increase. 



Keywords: Composite Cspar; Flutter speed; Aeroelastic instability; Aircraft tail; Airfoil section. 


© 2024 The Author(s). Mechanics of Advanced Composite Structures published by Semnan University Press. This is an open access article under the CCBY 4.0 license. (https://creativecommons.org/licenses/by/4.0/) 
The aerodynamic loads on the wings, tails, and blades are often tolerated by the main spar structure, which is manufactured using laminated composite materials. Nowadays, Cspars are used in the tails to withstand the loads and moments. The spar web is made of laminated composite material with a high degree of multiaxial layups, and the spar flanges contain a high number of unidirectional laminates. The special airfoil section shape creates lift and drag aerodynamic loads, while bending and torsional moments are borne by the spars and skin, respectively. The shape, materials, layups, position, and direction of the spar have decisive effects on the performance, vibration, and static/dynamic behaviors of the wings, tails, and blades. The mutual interaction of aerodynamic, structural, and inertial forces (dynamic aeroelasticity) can cause catastrophic failure of the aircraft, known as flutter phenomena. As the wind speed increases in an aircraft, there may be a point at which the structural damping is insufficient to dampen out the increasing motions caused by the addition of aerodynamic energy to the structure. This vibration can lead to structural failure, and therefore, considering flutter characteristics is an essential part of designing aircraft parts.
Numerical, analytical, and experimental studies for better perception of the flutter phenomena of the aircraft tail have been presented in recent years [1]. Optimized the wing structure with aeroelastic constraints [23], and aeroelasticity tailored and behaviors of the wing box and blades [45] are some of the analytical and numerical investigations for a better understanding of this phenomenon in the blades. A number of studies focused on the effects of the initial inplane and outofplane curvature [68] on the aeroelasticity instability (flutter) velocity, and some others used aviation standards for design and distinguish flutter speed point of the blades and wings [911]. The aeroelasticity (flutter) analysis of swept aircraft wings [1213] shows that concentrated mass has a complicated influence on the flutter boundaries.
The unsteady air loads affect the aeroelastic analyses using the doublet lattice method investigated by Van Zyl and Mathews [14]. The computational aeroelasticity methods for analysis of Ttail stability (flutter) boundaries for a freeflying aircraft in a transonic regime using a CFD formulation investigated by Attorni et al. [15]. Yu and Hu [16] have used an ultrasonic motor as an actuator in active flutter control of the airfoil section in the wind tunnel test. The aeroelasticity response of a bending–torsional coupling of wind turbine blade section with three degrees of freedom has been investigated by Stäblein et al. [15]. The structural stiffness of the airfoil section has assumed linear and bending–torsional coupling considered by the coupling coefficient in the stiffness matrix in their research [17]. Li and Ekici [18] using a oneshot method showed that the flutter boundaries can be very accurately predicted by prescribing a very small pitching amplitude.
Tang and Dowell [19] developed a numerical code for aeroelasticity analysis of the horizontal tail and mentioned that for a horizontal tail with flexible bending and torsion stiffness, a stall aerodynamic model and wing structural nonlinearity need to be considered. Zaki et al. [20] optimized the laminated composite arrangements to achieve maximum flutter with a minimum weight penalty. This research was one of the few studies in which the wing consisted of two spars, and the location of the spars was considered in the wing chord length for flutter investigation. Latif et al. [21] showed that changing the spar thickness contributes most significantly to the flutter speeds, whereas increasing the rib thickness decreases the flutter speed at high thickness values. Kumar et al. [22] modeled the geometric, structural, and aerodynamic parameters of the airfoil as independent variables. Furthermore, a review of the flutter of Ttail configurations [1] and a review of the nonlinear aeroelasticity of high aspectratio wings [23] showed that further work still needs to be conducted to identify aeroelastic instabilities that may occur in the tails and control surfaces.
Farsadi and Javanshir conducted a calculation of Flutter and Dynamic Behavior of Advanced Composite Swept Wings with Tapered crosssections in Unsteady Incompressible Flow [24]. Touraj Farsadi et al. [25] studied the geometrically nonlinear aeroelastic behavior of pretwisted composite wings modeled as thinwalled beams. Prasant Kumar Swain et al. [26] performed an aeroelastic analysis of a laminated composite plate with material uncertainty. Narayan Sharma et al. [27] conducted a stochastic frequency analysis of a laminated composite plate with curvilinear fiber. Narayan Sharma et al. [28] performed a free vibration analysis of a functionally graded porous plate using a 3D degenerated shell element. Amin Gharaei et al. [29] proposed an analytical approach for the aeroelastic analysis of tail flutter.
The literature review of the flutter phenomena, especially the flutter speed of the aircraft tail, shows that many of the research studies modeled the airfoil skin and did not concentrate on the effects of internal components such as the spar. This may be because of the importance of the skin in torsional frequency, but the effect of the spar must also be considered. In this research, the aeroelastic behavior, including flutter and divergence speed, of an aircraft tail section consisting of Cspars, is investigated. The main purpose of this manuscript is to study the influence of the different locations and directions of the Cspar. The elastic section modulus (S=I/c) and the mass of the laminated composite spar (M_s) are considered constant. Geometrical and physical parameters are determined using the ANSYS commercial software, and a twodegreeoffreedom classical flutter analysis method is used in the analytical process. Furthermore, by changing the position of the Cspar from the leading edge of the airfoil to the midchord, and also by changing the direction of the spar toward and backward to the leading edge, the aerodynamic instability, including flutter and divergence speeds of the wing, is analyzed.
In the analysis of aeroelastic flutter, an aerodynamic model (incompressible 2D subsonic model) is developed to capture the complex airflow using computational fluid dynamics or wind tunnel testing. The structural model incorporates the mechanical properties of the aircraft tail. The solution to aeroelastic instability involves coupling the aerodynamic and structural models and employing numerical methods such as computational structural dynamics. By analyzing the coupled system under various conditions, critical flutter speeds and the stability of the tail can be determined. It is important to note that the specific models and numerical techniques employed may vary based on the complexity of the problem and available resources, with computational aeroelasticity being an advanced approach for capturing the intricate interactions between aerodynamics and structural dynamics.
In this research, the aeroelastic behavior of a Vtail aircraft with a Cspar has been investigated. The airfoil section with the position (a) and direction (b) of the Cspar beam is shown in Figure 1. The common position of the spar is changed from 10% to 50% of the chord length from the leading edge of the airfoil section, as shown in Figure 1(a). The direction of the Cspar, represented by the orientation of the spar towards the left hand (LH) and backward right hand (RH) to the leading edge, can be seen in Figure 1(b).

a: The position of the Cspar from the leading edge 

b: Direction of the spar toward (LH) and 
Fig. 1. The direction and position of the laminated composite Cspar from the leading edge 
In the airfoil section with two degrees of freedom, as shown in Fig.2, the geometrical parameters are detected. The chord length is C and the half of the chord is determined as in this Fig.. The points AC, SC, and CG refer to the aerodynamic center, shear center(elastic center) or reference point in the flutter analysis, and the center of gravity, respectively. The location of the points SC and CG are determined using dimensionless parameters and with respect to the midchord, as shown in Fig.2, which . When these parameters are zero, the point lies on the midchord, and when they are positive/negative, the points lie toward the trailing/leading edges. In the structural design process, the skin is designed with 4 composite layers with 45degree layups. These layups will be constant in all cases.
Fig. 2. Tail section with two degrees of freedom and bending/torsional spring stiffnesses
The wing structural bending and torsional stiffnesses are modeled using discrete linear springs as shown in Fig. 2 [67, 30]. In the case of a Vtail with a smooth taper in the aircraft, the length of the mean chord at the midlength of the tail is chosen for analysis. The bending and torsional stiffnesses are then determined based on the material properties and thickness of the skin.
For the flutter analysis of the wing, calculation of torsional stiffness and bending stiffness is necessary. Bending stiffness was determined by equation1 as follows [31]:

(1) 
in the above equation, E, I_{x}, and L are Young’s modulus, a moment of inertia about the xaxis and length of the wing, respectively.
To determine the torsional stiffness, equation 2 is used [31]. In this equation, torsional stiffness is calculated by dividing the torsion of the wing, T by the torsional angle, .

(2) 
The torsional angle of the wing without spar found by equation3 [31]:

(3) 
A, G, and q are denoted to the airfoil area, shear modulus, and shear flow of the wing respectively. The is circumferential segment lengths and t is the skin thickness.
Since the wing with spar is divided into two cells and each cell has a torsional angle, these angles must be equal to ensure structural integrity. Equation 4 is used to determine the angles of torsion in each cell [31].

(4) 
the terms , represent summations around the entire perimeters of cell 1 and cell 2, respectively. indicates the value of the interior web and these terms are calculated by equation 5:

(5) 
The terms q1 and q2 are determined by equation6 as follows [29]:

(6) 
By substituting the previously mentioned values into equation 2, the torsional stiffness can be determined. Similarly, the bending stiffness can be obtained using Equation 1. It's important to note that the equations mentioned are applicable to isotropic materials. In the case of orthotropic materials, an equal modulus approach must be used.
To find them, an orthotropic transformed compliance matrix by φ angle ply for spar and skin layups must be evaluated. By and from compliance matrix, tension, and shear modulus ( ) can be found in equation 7 and equation 8. Substituting to equation1 and equation2, the required stiffness will be found [30].

(7) 

(8) 
The natural bending and torsional frequency are found by equation9 [22]:


(9) 
To simplify the problem, we consider the following four dimensionless variables as follows [30]:
(10) 


where r, σ, μ, and U are the dimensionless radius of gyration of the airfoil section about the shear center SC, the ratio of the uncoupled bending to torsional frequencies, the model mass to the mass of the air affected by the model, and reduced velocity as the dimensionless free stream speed of the air, respectively.
From Lagrange equations, the aeroelastic stiffness matrix can be written, that the determinant of this matrix must be equal to zero (equation11)
_{ } 
(11) 
There are two complex conjugate pairs of roots, that . For a given airfoil section, the behavior of the real and imaginary parts of the complex roots as functions of U will be calculated and discussed [30]. In the next section, this formulation has been applied to a Vtail hollow airfoil section and a section with spar. The divergence and flutter speeds are investigated for different spar locations. The obtained results for this specific airfoil section are discussed and analyzed.
In this research, various configurations of spars are investigated. The suggested thickness of the skin is 1.0 mm. The chord length of the airfoil is C = 431 mm, and the aerodynamic parameters are determined and presented in Table 1 using the ANSYS commercial software. To determine the bending and torsional stiffnesses using the finite element method, a new approach is employed. The flexural wing section value is defined as the ratio of the force applied at the bending section to the deflection of that section relative to the root section. Similarly, the torsional stiffness of the wing is calculated as the ratio of the moment to the angular rotation of the airfoil section. The torsional moment is produced by a pair of vertical forces applied to a frame fixed at the measuring section [31].
Mechanical properties of the unidirectional and woven carbon fiber reinforced polymers (CFRP) that are used in the tail, have been mentioned in Table 2. The E, ν, and ρ denoted the elastic modulus, Poisson's ratio, and density, respectively; and subscripts 1, 2, and 12 denoted the onaxis directions of the laminated composite materials.
The bidirectional (BD) woven carbon fiber reinforced polymers (CFRP) with ±45° layups are used in skin and sparweb. In the spar cap, unidirectional (UD) and ±45° BD are used, simultaneously. The difference among cap layups is because of preventing from changing elastic section modulus (S=I/c). Also, cap width changes to be constant in the elastic section modulus S.
Table 1. Geometric dimensions of wing with and without spar, based on different spar locations
Spar location and direction 
C.G 
S.C 
A (mm2) 
I_{SC } (m4 ) 

without Spar 
208.0 
116.3 
0.84E03 
3.20E02 
0.034 
0.460 
0.508 
7.605 
10% RH* 
166.7 
70.156 
1.15E03 
4.59E02 
0.226 
0.674 
0.538 
10.272 
10% LH** 
166.7 
102.8 
1.13E03 
3.61E02 
0.226 
0.523 
0.419 
10.393 
20% RH 
181.1 
89.0 
1.13E03 
4.01E02 
0.160 
0.587 
0.478 
10.108 
20% LH 
182.2 
126.3 
1.13E03 
3.01E02 
0.154 
0.414 
0.357 
10.162 
30% RH 
192.5 
116.2 
1.13E03 
3.28E02 
0.107 
0.461 
0.390 
10.134 
30% LH 
194.4 
157.6 
1.13E03 
2.44E02 
0.098 
0.268 
0.291 
10.108 
40% RH 
202.7 
145.8 
1.13E03 
2.68E02 
0.059 
0.323 
0.319 
10.154 
40% LH 
204.3 
185.0 
1.13E03 
2.16E02 
0.052 
0.142 
0.257 
10.110 
50% RH 
215.1 
169.6 
1.17E03 
2.49E02 
0.002 
0.213 
0.286 
10.506 
50% LH 
216.4 
209.9 
1.16E03 
2.13E02 
0.004 
0.026 
0.346 
10.412 
* Right hand side ** Left hand side 
Table 2. Mechanical properties of CFRP laminated composite materials
Materials 
E_{1} (GPa) 
E_{2} (GPa) 
G_{12} (GPa) 
ν_{12} 
ρ (kg/m^{3}) 
CFRP Unidirectional 
87.5 
7.5 
5.5 
0.28 
1600 
CFRP Bidirectional 
48 
48 
5 
0.05 
1600 
In Fig. 3, a twodimensional reference wing section with a spar is depicted. Bending and torsional loading were applied to determine the bending and torsional stiffnesses of the tail [3334]. For the bending stiffness, the value of the tail stiffness is defined as the ratio of the applied force at the bending section to the deflection ( ) relative to the root, as illustrated in Fig. 3(b). Similarly, for the torsional stiffness, the value is determined as the ratio of the bending moment applied at the root section to the angle of rotation ( ) of the reference section relative to the root section, as shown in Fig. 3(c). Therefore, the bending/torsional stiffnesses can be written as:


(12) 
In the finite element simulation depicted in Fig. 3, various spar locations and directions were considered, and the bending and torsional stiffnesses were obtained. This method, which involves using finite element analysis to determine the geometric properties and stiffnesses of the section, is referred to as the numerical method. In contrast, the analytical solution for these properties and stiffnesses is discussed in Section 3 of the research.
(a) 

(b) 

Fig. 3. Finite element simulation analysis to obtain a) Bending stiffnesses b) Torsional stiffnesses analyze
Table 3. Bending and torsional characterization (from theory) of airfoil sections for various spar configurations
Spar 
cap spar 
(N/m) 
(N.m/rad) 
Ωh 
Ωθ 
σ 

Theory 
FEM 
Δ 
Theory 
FEM 
Δ 

No Spar 
(for Skin) 
11280 
11,863 
4.9% 
16322 
16,318 
0.0% 
91 
714 
0.128 

10% RH 

25626 
25,832 
0.8% 
17210 
16,448 
4.6% 
118 
612 
0.193 

10% LH 

25626 
26,746 
4.2% 
17417 
16,713 
4.2% 
117 
694 
0.169 

20% RH 

28637 
30,503 
6.1% 
17455 
16139 
8.2% 
126 
660 
0.191 

20% LH 

28637 
31,379 
8.7% 
17674 
16688 
5.9% 
126 
767 
0.164 

30% RH 

29485 
30,395 
3.0% 
17779 
16099 
10.4% 
128 
736 
0.173 

30% LH 

29484 
30,805 
4.3% 
18135 
16978 
6.8% 
128 
863 
0.148 

40% RH 

28903 
29,985 
3.6% 
18250 
16506 
10.6% 
126 
825 
0.153 

40% LH 

28903 
29,811 
3.1% 
18642 
17652 
5.6% 
127 
930 
0.136 

50% RH 

26158 
27,128 
3.6% 
18756 
17283 
8.5% 
118 
867 
0.136 

50% LH 

26158 
24,001 
9.0% 
19102 
19625 
2.7% 
119 
948 
0.125 

Finite element models involve 40040 S4R elements. Bending and torsion loads are applied to each side of the tail symmetrically as boundary conditions. The static general finite element solver is employed for this purpose. A mesh study is conducted by using a mesh size ranging from 2 to 10 millimeters and results indicate that the mesh size has a minimal impact on the stresses with less than 8 percent.
The stiffnesses of the wing can be calculated analytically. Using this analytical approach, the frequencies and the ratio of the uncoupled bending to torsional frequencies have been determined for various types of sections. These results, obtained through the analytical method, are presented in Table 3. In addition to the analytical results, the finite element simulation results obtained using commercial software are also included in Table 3. A comparison is made between the numerical results and the analytical results, and the differences between these two methods are mentioned. It is stated that the observed differences between the results of these two methods are acceptable and within an acceptable range.
In the flutter analysis of orthotropic airfoil sections, the goal is to compare different spar configurations, locations, and positions. The steadystate method and thin airfoil theory for two degrees of freedom are utilized for this purpose. The flutter speeds in incompressible flow for different configurations are calculated and plotted using the governing equations. In this section, a specific section of CFRP (Carbon Fiber Reinforced Polymer) Vtail is considered for the flutter analysis. By substituting the CFRP parameters into the governing equations, the behavior of the complex roots as functions of the flight speed (V) is investigated to determine the divergent and flutter speeds. The imaginary part of the modal frequencies solution provides the primary graph of the flutter speed. The flutter point is located at the intersection of the torsional and bending (plunging) modal frequencies, where they coalesce. This point signifies the critical speed at which flutter instability occurs in the airfoil section.
By substituting the mechanical properties of CFRP, as mentioned in Table 3, into the governing equations for different spar positions and locations as specified in Table 4, the divergences and flutter speeds of the composite sections are obtained. The plots of the imaginary parts of the roots versus dimensionless speed for various spar configurations, as well as without a spar, are shown in Figure 4 and Fig. 5. In these figures, it can be observed that the flutter speed occurs at the intersection between pitching and plunging oscillations, while the divergence occurs at zero frequency. As the distance from the leading edge increases, both the dimensionless divergence and the flutter speed decrease. These figures provide a visual representation of the effects of spar positions and locations on the stability characteristics of the composite sections. It demonstrates how different configurations and the absence of a spar can impact the occurrence of divergence and flutter.

Fig. 4. The imaginary part of the modal solution, indicates the modal frequency versus V for the right spar web
Fig. 5. The imaginary part of the modal solution, indicates the modal frequency versus V for the left spar web
Table 4. Flutter and divergence speed (Km/h) for various spar cap and web locations
NO. 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
cap 
 
10% 
20% 
30% 
40% 
50% 

web 
 
RH 
LH 
RH 
LH 
RH 
LH 
RH 
LH 
RH 
LH 
VF 
1030 
1245 
1293 
1157 
1189 
1143 
1165 
1100 
1154 
1070 
1133 
VD 
1241 
1834 
1767 
1608 
1546 
1493 
1392 
1356 
1298 
1265 
1206 
In Fig. 5, as the distance from the leading edge increases, the dimensionless divergence and flutter speed decrease. This means that with a greater distance from the leading edge, the airfoil section becomes more stable, exhibiting a higher resistance to divergence and flutter. Regarding the conversion of dimensionless speeds to real speeds, multiplying the dimensionless speeds by the product (where b represents a characteristic length and ω_{θ} represents the dimensional angular frequency) can provide an estimation of the real speeds. However, it is important to note that the behavior of dimensionless speeds may not directly correspond to the behavior of real speeds. The dimensionless speeds, when plotted, may exhibit a rational distance among them, allowing for easier comparison and analysis. However, this behavior may not be directly observed in the real speeds. The relationship between dimensionless speeds and real speeds is influenced by various complex factors, including aerodynamics and structural characteristics. Therefore, it is crucial to consider the dimensional values and carefully analyze the specific context and factors affecting the airfoil's behavior when interpreting and comparing dimensionless and real speeds.
Except for the Mach number, the effective parameters in flutter speed are x_{θ}, r, σ, and μ.

(13) 
The frequency ratio (σ) is indeed influenced by the torsional frequency. In strength structural design, a straight wing is often assumed to behave like a beam subjected to aerodynamic bending loads. The spar is designed to withstand these aerodynamic loads while maintaining constant flexural stiffness. When changing the spar position, it is important to keep the flexural stiffness constant. This is achieved by maintaining a constant elastic section modulus (S=I/c) in all sections with a spar. As a result, the values of kh (the flexural stiffness factor) and ωh (the bending frequency) do not change.
Fig. 6 The imaginary Effective parameters versus spar position
However, moving the spar will affect the torsional rigidity. As the torsional frequency (ω_{θ}) increases, the frequency ratio (σ) decreases. This is because σ is defined as the ratio of the torsional frequency to the bending frequency. There are other parameters that can influence flutter, and these are plotted in Fig. 6. The position of the spar affects x_{θ}, which is the main factor influencing the inertia coupling stiffness (I_{SC}). Consequently, ISC affects r^{2} (the square of the radius of gyration) and σ. It is important to consider these factors and their interactions when designing and analyzing the wing structure to ensure stability and prevent flutter.
In Fig. 7, as the spar is moved from the leading edge to the trailing edge, several observations can be made:
Regarding the design requirements, such as those outlined in the FAR23.629 standard, it is recommended to have a margin of at least 20% between the maximum aircraft design speed and the flutter speed. Additionally, it is common in design references to consider a minimum margin of 1.5 in the ratio between torsional and bending frequencies. In this research, with a 20% margin from the flutter speed, all the ratios are above 3, indicating a satisfactory design margin in accordance with these criteria.
It is crucial to carefully analyze the dimensional values and consider the specific design requirements and standards when interpreting the results and making design decisions to ensure the stability and safety of the wing structure.
Fig. 7 The flutter and speed versus the spar position
In strength structural design, the sizing of spar caps is an important consideration. The direction of the Cspar (the direction of the spar caps) may depend on the manufacturing process, but in structural design, it is typically not a critical factor. The position of the spar web (the vertical member connecting the spar caps) will affect the size of the torsional cell and shear flows within the structure. However, it does not directly impact the sizing of the skin thickness, which is primarily determined by buckling considerations. It is important for the designer to be cautious because the position of the spar web can significantly influence the aeroelastic characteristics of the wing structure. In this research, the effect of the spar web position is investigated. In Fig. 8, when the spar web is moved from the lefthand direction to the righthand direction, the distance between the center of gravity and the elastic center will be considerably increased. This leads to an increase in the moment of inertia (ISC), which in turn decreases the torsional frequency (ωθ) and the frequency ratio (σ). Ultimately, this increase in moment of inertia and decrease in frequency ratio leads to an increase in the flutter speed (as shown in Fig. 6 and Fig. 8). These findings highlight the importance of considering the position and direction of the spar web in aeroelasticity considerations. The position of the spar web can have a significant impact on the torsional characteristics of the wing structure, affecting its stability and flutter behavior.
Fig. 8 Spars in one position (40%c) with different directions (different web positions)
For validation of the method, an old approximate equation is used based on the bending–torsional coupling [34] that refers to NACA RM L7G02. It is rewritten according to this research symbols as follows:

(14) 
where is the ratio of the mass of air (for diameter cylinder equal to the chord of the wing) to the mass of the wing, and both of them are taken for the length equals the span of the wing. Also, is the ratio of the massradius of gyration referred to shear center to the halfchord of the airfoil section.
Table 5. The discrepancy of flutter speeds for an aluminum section between results obtained from this research and the primary equation (Eq. 13)
Skin thickness (mm) 
0.5 
1 
1.5 
2 
Flutter speed (km/h) 
928 
1310 
1603 
1853 
Flutter speed (km/h) (NACA RM L7G02) 
858 
1210 
1471 
1686 
Discrepancy 
7.5% 
7.6% 
8.2% 
9.0% 
The calculated flutter speed obtained in this research can be compared to the results obtained from Eq. 14. These results for an aluminum section for various thicknesses based on primary Eq. 14 and results of the couplingtorsional coupling in this research are presented in Table 5. These results show that this approach has good agreement with other methods, and the percent discrepancy between these two methods is less than 10 %. As the primary Eq. 14 is based on torsional stiffnesses and disregarding the bending stiffnesses, it seems that this discrepancy between results is not out of reality and is acceptable for validating this research results.
Based on the results and conclusions obtained from the study on the Vtail aircraft using the steady method in incompressible flow, the following key findings were observed:
These conclusions highlight the importance of spar configuration and positioning in the aeroelastic behavior of the Vtail aircraft, providing valuable insights for structural designers to enhance stability and prevent flutter.
Overall, this article presents an applicable problem for aircraft designers who need to achieve standard levels for the structure to meet flutter FAR standard permissions.
This research did not receive any specific grant from funding agencies in the public, commercial, or notforprofit sectors.
The author declares that there is no conflict of interest regarding the publication of this article.
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