2.1 Geometrical configuration

This study examines the aeroelastic behaviour of a composite flat plate through two primary configurations. The first configuration serves as the baseline, consisting of a plate without stringers as shown in Fig. 1. Three different sweep angles, ${\rm{\Lambda }}$ , are considered (unswept, 25° Fw sweep, and 25° Aft sweep). Regardless of the sweep angle, the aerodynamic span is 305mm. The mass of the multi-layered unswept plate is approximately 0.0284kg. The fibre angle convention used is defined as 0° fibre when parallel to the flow velocity vector (x-axis) from the leading edge to the trailing edge, and 90° fibre travels along the semispan (y-axis) from root to tip. Throughout the entire work, only the first and last layers, theta, $\theta $ , of the laminate are varied in the sequence of ${[\theta, {45^ \circ }, - {45^ \circ }]_s}$ . This selection is made since the first and last layers are the furthest from the neutral axis, thus they contribute the most to bending rigidity.

The graphite/epoxy plate material and geometrical properties are taken from Refs. [Reference Hollowell and Dugundji18, Reference Stanford and Jutte19]. Table 1 summarises the laminate properties.

Table 1. Laminate properties [Reference Stanford and Jutte19]

Figure 1. Baseline plate geometry and fibre convention.

In the second configuration, an unswept, stiffened plate is considered. This involves using I and T-shaped stringers to enhance the stiffness of the plate. Initially, the stringers are placed parallel to the structural reference axis of the plate. Then, the orientation of the stringers is modified by introducing an angle between the stringers and the structural reference axis of the plate, referred to as swept stringers. In both cases, the spacing between the stringers is kept constant. This means that the distance between stringers remains unchanged regardless of whether the stringers are swept or unswept relative to the plate. Additionally, the mass of the plate is consistent for each cross-section of the stringers, regardless of their orientation (swept or unswept). The stringers are made from aluminium (AL 2024-T3) whose properties are taken from Ref. [Reference Gall and Boyer20]. Two different cross-sections are used as shown in Fig. 2, both with the same height, length and thickness.

Figure 2. (a) I-shape stringers (b) T-shaped stringers.

Figure 3 illustrates the unswept composite stiffened plate, where the stringers are placed parallel to the plate’s structural reference axis. To ensure consistency in the total mass of the stiffened plate and maintain the stringer spacing for both the unswept and swept stringer cases, a modification is made specifically for the unswept stringers case. In this modification, the cross-section of the trailing edge and leading-edge stringers is halved compared to the actual stringer cross-section as shown in Fig. 4 for a plate stiffened with I-shaped stringers.

Figure 3. Unswept stiffened plate.

Figure 4. A side view of the composite plate stiffened with I-shaped stringers.

Figure 5 shows the unswept stiffened plate with swept stringers. Table 2 illustrate the length of each swept stringer on the plate. The swept stringers are distributed with equal spacing around the centre of gravity (CG) so that the locus of CG is kept fixed. However, each strip (airfoil) of the plate has its own CG.

2.2 Flexural stiffness of the graphite/epoxy plate

${\bf{ABD}} \in {\mathbb{R}^{6 \times 6}}$ relates the applied load with the associated strains in the laminate. A, B and D are the extensional stiffness, bending-extension coupling, and flexural stiffness matrices, respectively.

(1) \begin{align}\left\{ {\begin{array}{*{20}{l}}{\bf{N}}\\{\bf{M}}\end{array}} \right\} = \left[ {\begin{array}{c@{\quad}c}{\bf{A}} & {}{\bf{B}}\\{\bf{B}} & {}{\bf{D}}\end{array}} \right]\left\{ {\begin{array}{*{20}{l}}\varepsilon \\\kappa \end{array}} \right\}\end{align}

where N is the vector of in-plane loads, M is the vector of bending/twisting moments, and $\varepsilon $ and $\kappa $ are the resulting mid-plane strains and curvatures, respectively. From an aeroelastic perspective, composite-based tailoring can positively or negatively alter the coupling terms in the ABD. The laminate stacking sequence is one of the main design drivers. The coupling terms ${{\rm{B}}_{16}}$ (bending-extension) and ${{\rm{B}}_{26}}$ (torsion-extension), which can be found in B, are an example of undesirable coupling, where the transverse loads along a wing cause both typical bending curvatures and an atypical in-plane extension, which can be large and nonlinear [Reference Stanford and Jutte19]. Nevertheless, in this paper, a symmetric staking sequence was used to eliminate the possibility of having any unfavourable coupling caused by B. Altering D coupling terms ( ${{\rm{D}}_{16}}$ ) and ( ${{\rm{D}}_{26}}$ ) is the most well-known and used strategy for tailoring by composites. The flexural modulus ${{\rm{D}}_{ij}}$ for n-ply laminate with arbitrary ply angle orientation can be obtained using:

(2) \begin{align}{{\rm{D}}_{ij}} = {1 \over 3}\mathop {\mathop \sum \limits_{k = 1} }\limits^n {\rm{Q}}_{ij}^{\left( {{\theta _k}} \right)}\left[ {z_k^3 - z_{k - 1}^3} \right]{\rm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}}ij = 1,2...6\end{align}

where ${\rm{Q}}_{ij}^{\left( {{\theta _k}} \right)}$ is the off-axis lamina modulus of the ${k^{{\rm{th}}}}$ ply, ${\theta _k}$ is the ply angle of the ${k^{{\rm{th}}}}$ ply and ${z_k}$ is the vertical distance from the midplane to the upper surface of the ${k^{{\rm{th}}}}$ ply [Reference Hollowell and Dugundji18]. ${{\rm{D}}_{ij}}$ can be written in a matrix form as:

(3) \begin{align}{\bf{D}} = \left[ {\begin{array}{c@{\quad}c@{\quad}c}{{{\bf{D}}_{11}}} & {}{{{\bf{D}}_{12}}}& {}{{{\bf{D}}_{16}}}\\{{{\bf{D}}_{12}}}& {}{{{\bf{D}}_{22}}}& {}{{{\bf{D}}_{26}}}\\{{{\bf{D}}_{16}}}& {}{{{\bf{D}}_{26}}}& {}{{{\bf{D}}_{66}}}\end{array}} \right]\end{align}

The flexural moduli components ${{\rm{D}}_{ij}}$ for the stacking sequence used in this work ( ${[\theta, {45^ \circ }, - {45^ \circ }]_s}$ ) on the entire range of theta ( $ - {90^ \circ } \lt \theta \lt {90^ \circ }$ ) were computed using Nastran and plotted in Fig. 6.

Table 2. Length of each swept stringer illustrated in Fig. 4

Figure 5. Unswept stiffened plate with swept stringers. The figure illustrates two flow directions; one to analyse the effect of sweeping the stringers Aft (positive $\theta $ ), and the other when sweeping them Fw (negative $\theta $ ).

Figure 6. Flexural moduli components for $ - {90^ \circ } \lt \theta \lt {90^ \circ }$ .

2.3 Structural model

The plate is modelled in Nastran using laminate elements for structural representation. A clamped boundary condition is applied at the root of the plate using nodal constraints. A structural mesh of (36×144) is selected after conducting a mesh convergence analysis where the natural frequencies of the first ten modes are determined for different mesh sizes. Figure 7, where n represents the multiple of 12 chordwise and 48 spanwise elements (12n×48n), respectively, concludes that the analysis is completely mesh-insensitive.

Figure 7. Mesh sensitivity analysis.

2.4 Aerodynamic model

Aerodynamic elements are regions of lifting surfaces and are represented using DLM. The rectangular aerodynamic coordinate system defines the flow direction as positive along the x-axis. The aerodynamic mesh is composed of 24×96 elements. DLM is based on the concept of representing a lifting surface as a combination of doublets. By discretising the lifting surface into a lattice of doublets, the aerodynamic characteristics of the surface can be analysed and calculated [Reference Albano and Rodden21].

2.5 Aeroelastic equations of motion

In general, the equations of motion used to describe the behaviour of an aeroelastic system can be expressed in a matrix form:

(4) \begin{align}\textbf{A}\ddot{\textbf{q}} + \left( {\rho V{\bf{B}} + {\bf{D}}} \right)\dot{\textbf{q}} + \left( {\rho {V^2}{\bf{C}} + {\bf{E}}} \right){\bf{q}} = {\bf{f}}\end{align}

where A is the structural inertia, B and D are the aerodynamic and structural damping respectively, and C and E are the aerodynamic and structural stiffness respectively. q is the generalised plunge and pitch degrees of freedom. f is the generalised forces. V and $\rho $ are the airspeed and air density, respectively [Reference Wright and Cooper22]. In Nastran, a flutter analysis, which determines the dynamic stability of an aeroelastic system, is conducted to find both flutter and divergence speeds. The speeds are found using the British PK method. The primary advantages of the PK method are twofold. Firstly, it directly yields results corresponding to specific velocity values, allowing for precise analysis. Secondly, it offers a reliable estimation of system damping at subcritical speeds, which can be utilised for monitoring flight flutter tests. In the PK method, the aerodynamic matrices are treated as real springs and dampers that depend on frequency. The method involves estimating a frequency and then finding the eigenvalues. From an eigenvalue, a new frequency is obtained. The input data for the PK method allows for looping, enabling the analysis to be performed iteratively. The inner loop contains the velocity set, while the outer loops consider Mach number and density values. This allows for the examination of the effects resulting from changes in one or both parameters within a single run.

Figure 8. Surface spline and its coordinates (adapted from Ref. [Reference Nastran23]).

2.6 Structure and aerodynamics interconnection (spline interpolation)

For aeroelastic analysis, the aerodynamic (dependent degrees of freedom) and structural meshes (independent degrees of freedom) are connected through splines as shown in Fig. 8. Equation (5) is used to relate the independent and dependent degrees of freedom. Through splining, two transformations are required: an interpolation from the structural to the aerodynamic deflections and a connection between the aerodynamic and the structurally equivalent forces acting on the structural grids.

(5) \begin{align}\left\{ {{u_K}} \right\} = \left[ {{G_{{\rm{kg}}}}} \right]\left\{ {{u_g}} \right\}\end{align}

where $\left[ {{G_{{\rm{kg}}}}} \right]$ is an interpolation matrix that relates the components of structural grid point deflections $\left\{ {{u_g}} \right\}$ to the deflections of the aerodynamic grid points $\left\{ {{u_K}} \right\}$ . The interpolation matrix depends on the type of spline used, (surface spline) for this work. The aerodynamic forces, denoted by $\left\{ {{F_k}} \right\}$ and their corresponding structurally equivalent values $\left\{ {{F_g}} \right\}$ apply identical virtual work on the structural grid points within their respective deflection modes.

(6) \begin{align}{\{ \delta {u_k}\} ^T}\left\{ {{F_k}} \right\} = {\{ \delta {u_g}\} ^T}\left\{ {{F_g}} \right\}\end{align}

where $\delta {u_k}$ and $\delta {u_g}$ are the aerodynamic and structural virtual deflections, respectively, substitute (5) into (6) and rearrange to get the required force transformation,

(7) \begin{align}\left\{ {{F_g}} \right\} = {[{G_{kg}}]^T}\left\{ {{F_k}} \right\}\end{align}

Equations (5) and (7) are both required to connect aerodynamics and structural grids for any aeroelastic problem [Reference Nastran23].

2.7 Validation of the baseline plate model

The baseline plate model is validated by conducting flutter analysis using Nastran. The resulting flutter, divergence and natural mode frequencies are compared to those obtained from an optimisation study referenced as Ref. [Reference Stanford and Jutte19], as well as the experimental and computational work in Ref. [Reference Hollowell and Dugundji18] and tabulated in Table 3.

Table 3. Baseline plate model validation

It should be noted that the fibre convention utilised in Ref. [Reference Hollowell and Dugundji18] was ${0^ \circ }$ spanwise, therefore, to compare their plate with the plate used in this work, the fibre orientation is set to ${0^ \circ }$ spanwise, and the stacking sequence is oriented ${90^ \circ }$ .

3.1 Effect of plate sweep angle and ply orientation (configuration 1)

The well-known behaviour related to the impact of sweep angle subjected to upwards bending states that; for the upswept configuration, the incidence angle remains unchanged, leading to pure deformation of the plate. For the Aft swept case, the washout effect causes a reduction in the effective streamwise angle of incidence. As a result, the divergence speed increases. Conversely, for the Fw swept case, the washin effect takes place, leading to an increase in the effective streamwise angle of incidence. Consequently, the divergence speed decreases [Reference Wright and Cooper22]. When composites are introduced, the alteration in the geometric tailoring effect due to the sweep angle can be balanced, increased or decreased.

Theta (ply angle) is varied for the range of $ - {90^ \circ } \lt \theta \lt {90^ \circ }$ with a step of ${5^ \circ }$ . As theta is the only design variable, the obtained results directly correspond to the coupling terms of the ABD, specifically the ${D_{26}}$ (spanwise bend-twist coupling) term. Figure 6 illustrates all D components across the entire theta range. Based on the sign of the bending-torsion coupling term, three distinct scenarios can be observed: 1- zero coupling terms, creating a pure bending deflection, 2- negative coupling terms (washout) and 3- positive coupling terms (washin).

In Fig. 9(a), the unswept plate exhibits two changes in the flutter mode: at $ - {50^ \circ }$ and ${65^ \circ }$ , both coinciding with peaks in flutter frequency. Within the range from $ - {90^ \circ }$ to $ - {50^ \circ }$ , the flutter occurs in the second mode, in this region the deformation type changes from first torsion (1T) to second bending (2B) around $ - {75^ \circ }$ , it can be observed from the frequency Fig. 9(b) that this transition is accompanied by a convergence between the second and third modes, as these modes approach each other in terms of their frequencies.

Figure 9. (a) Aeroelastic instability velocities vs theta (b) Flutter and the natural frequencies of the first four modes vs theta for unswept composite plate.

When the fibre orientation is modified, the primary stiffness axis shifts either Aft or Fw. By varying the fibre angle, it is possible that some fibres do not extend continuously from the root to the tip of the plate. This can lead to an unfavourable positioning of the primary stiffness axis. Furthermore, aligning the fibres at a specific angle, such as $ - {50^ \circ }$ , can be advantageous for both flutter and divergence. Aeroelastic instabilities reach their maximum values at this angle. However, it should be noted that at $ - {50^ \circ }$ , the fibre orientation introduces a washout effect, which is generally unfavourable for flutter. The peak in divergence speed observed between $ - {85^ \circ }$ and $ - {20^ \circ }$ is linked to negative coupling terms (washout). Nevertheless, increasing the divergence speed can be associated with a flutter speed reduction or improvement. Inversely, an increase in flutter may lead to a lower or higher divergence speed; therefore, more design variables are employed in the coming studies to find the best combination of structural and material that fulfils aeroelastic tailoring objectives.

As stated in the preceding section, the zero degrees fibre is oriented parallel to the velocity vector. In the case of a swept geometry, the convention remains fixed, but it should be noted that fibre orientation is shifted by ${25^ \circ }$ due to the geometrical sweep as illustrated in Fig. 10. Furthermore, along the span, if ${90^ \circ }$ is the angle at which the fibres are extended continuously from root to tip, the equivalent would be ${65^ \circ }$ on an Aft swept plate.

Figure 10. Fibre orientation for an unswept and Aft swept plate.

Figure 11 demonstrates the impact of sweeping the plate ${25^ \circ }$ Aft. It is observed that the divergence speed significantly increases compared to the unswept case. However, in the positive region of theta, a substantial decrease in divergence speed is observed. Despite the material’s positive coupling (washin) in this region, the Aft swept geometry (washout) still results in a higher divergence speed compared to the previous case. The figure also shows four changes in the flutter mode, each associated with peaks in flutter frequency. At ${50^ \circ }$ , the deformation type of the second mode changes from (1T) to (2B). Like the previous case, this change coincides with a convergence between the second and third modes as they approach each other in terms of their frequencies. The maximum flutter speed observed was 37m/s at ${55^ \circ }$ . This is roughly due to the arrangement of the fibres on the swept geometry around this angle. In this region, the fibres are extended continuously from the root to the tip of the plate. Predictably, giving the maximum washin effect on a ${25^ \circ }$ Aft swept geometry.

Figure 11. (a) Aeroelastic instability velocities vs theta (b) Flutter and the natural frequencies of the first four modes vs theta for ${25^ \circ }$ Aft swept composite plate.

Figure 12 shows that sweeping the plate ${25^ \circ }$ Fw results in an improvement in flutter speed when compared with the Aft sweep. However, as a consequence of this flutter increase, the divergence occurs at very low speeds, except in the region where the material exhibits a washout effect (in a sense; balancing the effect of geometry). Between $ - {75^ \circ }$ and $ - {45^ \circ }$ the flutter occurs in the second mode, and within this region, the deformation type changes from (1T) to (2B), where the modes approach each other in terms of their frequencies.

Figure 12. (a) Aeroelastic instability velocities vs theta (b) Flutter and the natural frequencies of the first four modes vs theta for ${25^ \circ }$ Fw swept composite plate.

Figure 13. Flutter speed vs theta (b) Flutter frequency vs theta for the unswept stiffened plate with I-shaped and T-shaped stringers.

In summary, a maximum flutter speed of 120m/s, which occurs in the third mode (2B), is achieved when sweeping the plate ${25^ \circ }$ Fw and orienting the fibres by $ - {75^ \circ }$ . On the other hand, the optimal divergence speed is 84m/s, which occurs in the first mode (1B), when sweeping the plate ${25^ \circ }$ Aft and orienting the fibres by $ - {80^ \circ }$ . These findings highlight the impact of both sweep direction and fibre orientation on the aeroelastic behaviour of the plate, with different combinations yielding distinct flutter and divergence characteristics.

3.2 Effect of stiffening the plate (configuration 2 with unswept stringers)

In this section, the focus is on investigating the influence of incorporating stiffeners into an unswept composite plate, as well as assessing the effects of altering the cross-sectional shape of the stringers. Both I and T-shaped stringers are considered. As shown in Figs. 13(a) and 14, the influence of theta on aeroelastic instabilities is relatively minor. In the negative region of theta, where the coupling term ( ${{\rm{D}}_{26}}$ ) is negative, and the material is giving a washout effect, a slight increase in aeroelastic instabilities is observed compared to the positive range of theta. This trend is similar to that of the unswept plate (unstiffened) where an increase in flutter and divergence speeds is achieved in the negative range of theta. However, the increase achieved in the negative range of theta for the unswept stiffened plate is reduced. This suggests that the variation in theta has a limited impact on the flutter and divergence characteristics of the stiffened plate mainly because the stiffeners are the main contributors to the rigidity of the structure. Moreover, the behaviour of the first and second modes as well as the flutter frequency (Fig. 13(b)) is almost independent of theta. In contrast to subsection (1) where changes in the flutter mode (between the 2nd and 3rd modes) are observed for different theta values, no changes are observed for the entire range of theta when stiffeners are added. Regardless of the stiffener’s cross-section, the flutter always occurs in the second mode (1T). This further supports the notion that the presence of stringers and the resulting increase in stiffness have a stabilising effect on the plate’s aeroelastic behaviour.

Figure 14. Divergence speed vs theta for the unswept stiffened plate with I-shaped and T-shaped stringers.

Both I and T-shaped stringers exhibit a comparable pattern in terms of aeroelastic behaviour. This similarity can be attributed to their relatively similar geometry and mass distribution near the attachment area with the plate. Figures 13 and 14 illustrate this trend, highlighting the influence of the stringer cross-section shape on aeroelastic performance. Moreover, the utilisation of T-shaped stringers results in the vertical positions of the centre of gravity and the effective neutral axis being closer to the plate, compared to I-shaped stringers. On the other hand, the additional flange in the I-shaped stringers shifts the centre of gravity and the neutral axis downwards, providing higher bending rigidity. This causes variations in mode shapes and natural frequencies.

3.3 Effect of varying the stringer’s sweep angle (configuration 2)

In this subsection, the influence of sweeping the stringers is examined. It is observed that sweeping the stringers alone has a different effect compared to sweeping the entire plate. Unlike sweeping the entire plate, where Aft sweep results in a washout effect and Fw sweep results in a washin effect, sweeping the stringers alone reverses this well-known behaviour. This observation highlights the importance of considering the specific configuration on a more realistic wing box model. Furthermore, this finding suggests that sweeping the stringers alone could potentially address the aeroelastic instabilities associated with Fw swept wings and provide a solution for such design considerations.

The effect of sweeping the stringers with respect to the applied load axis generates shear loads within the skin. For Fw swept stringers subjected to upward bending (from lift) results in a washout deformation. On the contrary, a plate with Aft swept stringers under upward bending (from lift) experiences a washin deformation. Figure 15 shows how the applied load is resolved into two components, one acting along the stringer’s direction and one normal to the stringer’s direction. The normal component is what causes the plate to twist (nose-down). The coupling effect may differ based on the amount of the stringer’s orientation angle [Reference Williams24, Reference Gimmestad25].

Figure 15. Loads redistribution along the swept stringers.

Figure 16. (a) Average tip displacement vs angle-of-attack (b) Tip twist vs angle-of-attack for configuration 2.

3.3.2 Flutter analysis

Aft swept stringers Fig. 17 illustrates the impact of varying the ply orientation, sweeping the stringers ${25^ \circ }$ Aft, and changing the stringer’s cross-section. Several conclusions can be drawn from Fig. 17(a). In the positive region of theta, where both the material and geometry contribute to a washin effect, the flutter speed is generally higher. This can be attributed to the combined effect of the favourable geometry and material properties. In the negative range of theta, changes in the flutter mode can be observed, this is roughly due to the contradicting contributions from the material which gives a washout effect, and geometry which contributes to a washin effect. Nevertheless, changes in flutter mode are associated with frequency peaks as shown in Fig. 17(b) and stressed previously.

Figure 17. (a) Flutter speed vs theta (b) Flutter frequency vs theta for the unswept stiffened plate with ${25^ \circ }$ Aft swept I-shaped and T-shaped stringers.

When I-shaped stringers are employed, Fig. 17(a) indicates two changes in the flutter mode between the third and second modes ((2B) and (1T), respectively). These mode changes can be attributed to different factors. The first change in flutter mode may be influenced by the transition in the material effect from washin to washout, as depicted in Fig. 6. The second change in flutter mode could be associated with the material effect as well. In this range, an intersection of the ${{\rm{D}}_{26}}$ and ${{\rm{D}}_{16}}$ terms occurs (Fig. 6). Similarly, when T-shaped stringers are employed, two changes in flutter mode can be observed around the same region. These mode changes and the shift in the dominant mode of flutter occur due to similar underlying assumptions, involving the interplay between material effects and geometric factors.

Figure 18 demonstrates the influence of the material effect on the divergence speed of the stiffened plate with Aft swept stringers. Specifically, in the negative range of theta, where the material exhibits a washout effect, an increase in divergence speed can be observed regardless of the stringer’s cross-section.

Figure 18. Divergence speed vs theta for the unswept stiffened plate with ${25^ \circ }$ Aft swept I-shaped and T-shaped stringers.

Fw swept stringers

Figure 19 illustrates the influence of sweeping the stringers Fw, changing the stringer’s cross-section, and varying the ply orientation on the aeroelastic behaviour of the stiffened plate. It can be observed that the flutter speed for both cross-sections is higher in the positive region of theta, where the material is giving a washin effect, thus balancing the washout effect caused by the geometry. Nevertheless, as concluded earlier, I-shaped stringers result in higher aeroelastic instabilities speeds compared to T-shaped stringers.

Figure 19. (a) Flutter speed vs theta (b) Flutter frequency vs theta for the unswept stiffened plate with ${25^ \circ }$ Fw swept I-shaped and T-shaped stringers.

Figure 20 illustrates the plate’s divergence speed, which is extremely high for both cross-sections. The I-shaped stringers result in a divergence speed above 360m/s, while the T-shaped stringers exhibited a divergence speed greater than 250m/s. The significant increase in the divergence occurs when both the material and geometry contribute to the washout effect. From Fig. 20, it can be shown that the divergence speed for the plate stiffened with I-shaped stringers cannot be detected in the negative range of theta, as it exceeds 1,100m/s. These high divergence speeds make the flutter velocity a critical consideration for this particular case. The washout effect caused by the geometry contributes to the high divergence velocities observed in this study.

Figure 20. Divergence speed vs theta for the unswept stiffened plate with ${25^ \circ }$ Fw swept I-shaped and T-shaped stringers.