Nomenclature
Roman symbols
- c
-
aerofoil chord
-
${c_l}$
-
lift coefficient per unit span
- E
-
Young’s modulus
-
${\textbf{f}_{{\rm{spoiler}}}}$
-
force vector on mesh due to spoiler
-
$H$
-
gust gradient, as defined in CS-25.341 [8]
-
${I_{ii}}$
-
second moment of area about the
$i$
-axis - kCAS
-
calibrated airspeed in knots
-
${\rm{M}}$
-
Mach number
-
${M_i}$
-
bending moment about the
$i$
-axis, positive in a right-hand screw sense -
${S_{ij}}$
-
shear force acting on the
$i$
face in the
$j$
direction -
${T_c}$
-
convective time unit
$ = c/{V_\infty }$
-
${V_{\rm{C}}}$
-
cruise airspeed
-
${V_{{\rm{MO}}}}$
-
maximum operating airspeed
-
${V_{{\rm{stall}}}}$
-
airspeed at stall
-
${V_\infty }$
-
true airspeed of freestream
-
$Z$
-
altitude
-
${Z_{{\rm{MO}}}}$
-
maximum operating altitude
Greek symbols
-
$\alpha $
-
angle-of-attack
-
${\rm{\Delta }}$
-
maximum possible spoiler deployment angle
-
$\delta $
-
spoiler deployment angle at a given instant
-
${\varepsilon _{ij}}$
-
strain at the
$i$
face in the
$j$
direction -
${\varepsilon _{{\rm{spoiler}}}}$
-
value of
${\varepsilon _{11}}$
at the spoiler location -
${\kappa _{ii}}$
-
beam curvature about the
$i$
-axis
1.0 Introduction
Gust load alleviation has long been a goal in the design of aircraft wings. Reducing the peak stresses caused by extreme gust events effectively increases the structural margin of a wing’s design. This additional margin can then be re-deployed by adopting a more aerodynamically efficient design, e.g. by increasing span, ultimately resulting in a more fuel-efficient aircraft design. As a result, many active load control systems have been proposed or implemented on aircraft wings [Reference Al-Battal, Cleaver and Gursul1, Reference Castrichini, Siddaramaiah, Calderon, Cooper, Wilson and Lemmens5, Reference Jeanneau, Aversa, Delannoy and Hockenhull12, Reference Regan and Jutte17, Reference Stalla, Kier, Looye, Michel, Schmidt, Hanke, Dillinger, Ritter and Tang23]. However, active load control systems have the disadvantage that they require external sensors and actuators to function, and this adds weight and complexity to an airframe.
A natural response to these drawbacks is to instead utilise passively actuated load alleviation devices, which do not rely on external systems. Instead, the wing structure is designed such that the intelligence traditionally provided by active components is ‘built into’ the wing’s structural behaviour. One now well-established embodiment of this philosophy is aeroelastic tailoring [Reference Shirk, Hertz and Weisshaar22]. More recently, the concept of ‘well-behaved nonlinear structures’ [Reference Champneys, Dodwell, Groh, Hunt, Neville, Pirrera, Sakhaei, Schenk and Wadee7, Reference Groh, Avitabile and Pirrera9, Reference Reis18] has been used to design passive load alleviation systems which exploit nonlinear structural phenomena [Reference Arrieta, Kuder, Rist, Waeber and Ermanni2, Reference Cavens, Chopra and Arrieta6, Reference Runkel, Reber, Molinari, Arrieta and Ermanni20, Reference Thel, Hahn, Haupt and Heimbs25]. Runkel et al. [Reference Runkel, Reber, Molinari, Arrieta and Ermanni20] designed and tested a wingbox whose rear spar web was designed to buckle above a certain load threshold. This buckling shifts the shear centre of the wingbox forward, causing a reduction in the wing’s angle-of-attack and thus providing a load alleviation benefit [Reference Hahn, Haupt, Lobitz and Heimbs11, Reference Runkel, Fasel, Molinari, Arrieta and Ermanni19]. The buckling spar relies on geometric nonlinearity in an underlying conservative structure to achieve passive load alleviation. Szczyglowski et al. [Reference Szczyglowski, Neild, Titurus, Jiang, Cooper and Coetzee24] proposed a passive load alleviation whose nonlinearity originates in a dissipative structural element. The researchers proposed a truss-braced wing concept in which discrete rotary dampers were introduced into the joints between the bracing strut and the rest of the airframe, allowing for reduced internal stresses in the wing.
Aeroelastic modelling of these concepts [Reference Hahn, Haupt, Lobitz and Heimbs11, Reference Runkel, Fasel, Molinari, Arrieta and Ermanni19, Reference Szczyglowski, Neild, Titurus, Jiang, Cooper and Coetzee24] has demonstrated that they can provide a load alleviation benefit. However, they also require a significant alteration to the wingbox design utilised in most modern commercial aircraft. An alternative is to take a more localised approach, whereby aerodynamic control surfaces are actuated passively in order to achieve a load alleviation benefit. A number of concepts have been proposed in which a section of the wing’s trailing edge is designed to buckle under increased aerodynamic load, locally reducing the wing’s lift coefficient [Reference Arrieta, Kuder, Rist, Waeber and Ermanni2, Reference Cavens, Chopra and Arrieta6, Reference Hahn, Haupt and Heimbs10]. Another localised scheme is to vent fluid from the lower surface to the upper surface of the wing [Reference Seki, Tani and Aso21], or to adapt existing control surfaces for passive actuation [Reference Lancelot and De Breuker15]. However, whilst these control surface based systems are less impactful on a wing’s architecture, they rely on local aerodynamic pressures as their primary stimulus for actuation. Aerodynamic pressures are clearly related to internal wingbox loads, but are not their direct proxy, and vary considerably throughout a flight envelope. As such, they may not be a suitable stimulus for activation of a gust load alleviation system.
In earlier work [Reference Wheatcroft, Shen, Groh, Pirrera and Schenk26, Reference Wheatcroft, Shen, Groh, Pirrera and Schenk28], we presented a passive gust load alleviation device which combines the benefits of the two approaches described above; the device consists of a spoiler control surface, which deploys in response to a critical strain in the connected wing structure. Deployment of the control surface is driven by a morphing nonlinear structure, which stores strain energy from the wingbox during the initial phase of a gust, before releasing it dynamically via a structural instability (i.e. buckling) when a critical level of input strain is exceeded. This energy is used to rapidly deploy a leading-edge tab, which stalls the flow over the wing and thus reduces the overall lift the wing experiences during the gust. The spoiler is then automatically retracted once input strain drops below a safe value, returning the wing’s aerodynamic profile to its clean configuration. Recent wind tunnel testing [Reference Wheatcroft, Mahadik, Groh, Pirrera and Schenk27] has shown that the prototype can rapidly and reliably reduce a wing’s lift coefficient upon deployment, and that its nonlinear structural behaviour is largely unaffected by the addition of aerodynamic loading. These results demonstrate that the prototype is capable of locally reducing the lift generated by a wing purely in response to strain. However, the modelling and experiments conducted thus far do not account for the more global dynamics and aeroelastics of an airframe subjected to gust loading, which clearly have a significant impact on the load alleviation benefit the passive spoiler can provide.
In the present work, we conduct numerical aeroelastic modelling of an airframe, with the aim of verifying that a strain-actuated spoiler is capable of providing a load alleviation benefit at aircraft scale. We conduct transient analyses in which an airframe model is subjected to a series of vertical One-Minus-Cosine (1MC) gusts. The strain at a given wing location is monitored during the analysis, and a downward spoiler load is applied to the wing whenever the strain meets a triggering condition. This behaviour is designed to closely mirror that of the aforementioned spoiler prototype [Reference Wheatcroft, Mahadik, Groh, Pirrera and Schenk27], capturing its effect on wing loads whilst avoiding the need to directly model its complex nonlinear dynamics. Wing root bending moment is monitored in order to assess the amount of load alleviation provided by the spoiler. This analysis not only demonstrates and quantifies the efficacy of a strain-actuated spoiler as a load alleviation device, but it allows the impact of key spoiler design parameters to be investigated. This information is likely to become an essential design tool when developing future iterations of the spoiler. The possibility that the spoiler induces limit cycle oscillations in the wing and means to mitigate this issue are also investigated. A preliminary investigation into the load alleviation performance of a strain-actuated spoiler was conducted by Castrichini et al. [Reference Castrichini, Consentino, Siotto, Sun, Coppin, Vargas, Ruiz and Hadjipantelis3, Reference Castrichini, Cosentino, Siotto, Sun, Coppin, Vargas, Ruiz and Hadjipantelis4]. Crucially, though, the spoiler in that study was activated in response to flexural displacement of the wing tip, which is not an unambiguous metric of strain at a given span station during dynamic deformations.
The remainder of this paper is structured as follows. Details of the aeroelastic model and analysis procedures are given in Section 2, as well as a discussion of the capabilities and limitations of the modelling approach. Results of the modelling are presented and discussed in Section 3. These are split into steady state results (Section 3.1) and transient results (Section 3.2). Section 4 concludes the paper.
2.0 Methodolgy
This section introduces the model and procedures used to analyse the passively actuated gust load alleviation spoiler. The capabilities and limitations of the modelling strategy are also discussed.
2.1 Model and analysis procedures
This work uses the Future Fast Aeroelastic Simulation Technologies (FFAST) aeroelastic model [Reference Jones, Gaitonde, Knörzer and Szodruch13, Reference Khodaparast and Cooper14]. This model was selected as it was readily available, and is representative of a modern wide-body commercial airliner. FFAST is a Finite Element (FE) model in MSC NASTRAN [16], which consists of connected structural and aerodynamic meshes, as illustrated in Fig. 1. The structural mesh is a ‘stick’ model, with the principal components of the airframe represented by NASTRAN CONM2 lumped masses connected by either flexible CBAR or rigid RBE2 elements. The stiffness of a few elements near the wing tip was increased from the original model, as the strains in these elements were unrealistically high. This also gave a more realistic uniform strain distribution throughout the wing span (c.f. Fig. 5(d)). The aerodynamic mesh consists of CAERO1 elements that represent the wings and the horizontal and vertical stabilisers. An elevator control surface is included in the trailing edge of the horizontal stabiliser. Forces and displacements are transferred between these meshes using SPLINEn cards. All degrees of freedom except for pitch and plunge are restrained at the aircraft’s centre-of-gravity (CG) node.
The NASTRAN model used in the analysis consists of (a) a structural mesh connected to (b) a mesh of aerodynamic panels. Each triangle denotes a structural node.
MSC NASTRAN’s static and dynamic aeroelastic analyses [16] are conducted in the frequency domain, which requires all external loads to be defined a priori. This is not possible here, because the spoiler must deploy in response to the wing strain, which is itself a function of the unknown deformed state. The analyses must therefore be conducted in the time domain. To achieve this, NASTRAN’s Direct Matrix Abstraction Program (DMAP) is used to extract the structural mass, stiffness and damping matrices of the model along with the generalised aerodynamic force matrices. These are then read into Python, where the model’s equations of motion are numerically integrated; full details of the method are given in Castrichini et al. [Reference Castrichini, Siddaramaiah, Calderon, Cooper, Wilson and Lemmens5]. The procedure recovers the deformed state at each time step, allowing computation of strains and thus spoiler loads during the analysis. The analysis uses the model’s vibration modes as basis functions for describing displacement degrees of freedom, so the model’s degrees of freedom become the amplitude of each modal basis function. In order to reduce run-time, only modes with a natural frequency of
$40$
Hz or less are incorporated. This is more than double the peak gust excitation frequency of approximatively
$15$
Hz (based on a minimum gust length of
$18$
m and a maximum true airspeed of
$276$
m/s).
The solution procedure involves two steps: a static aeroelastic trim analysis followed by a dynamic gust analysis. The trim solution is used as an initial condition for the dynamic analysis, in which the model is exposed to a 1MC gust as defined by the European Union Aviation Safety Agency in CS-25 [8]. Five gust gradients,
$H$
, equally spaced between the regulation minimum and maximum, were considered:
$9$
m,
$21.3$
m,
$45.7$
m,
$76.2$
m and
$107$
m. The principal excitation frequency associated with each gust gradient is given by
${V_\infty }/2H$
, although it should be noted that the frequency content of a 1MC signal spans a finite band [Reference Szczyglowski, Neild, Titurus, Jiang, Cooper and Coetzee24]. The gust analysis was conducted at each of the
$16$
different flight points shown in Fig. 2(a), which are intended to span the typical flight envelope of a modern commercial airliner. The gust frequency associated with every flight point and gust gradient is shown in Fig. 2(b). The frequency of the airframe’s first symmetric wing bending mode, equal to
$2.19$
Hz, is also shown for reference. The results of sample trim and dynamic analyses without the spoiler were compared to results from NASTRAN solution 144 and 146 analyses in order to verify the functionality of the Python script. Good agreement was achieved, giving confidence in the methodology. All structural analysis was conducted on the starboard wing for simplicity, since the structure is symmetric and subjected only to symmetric loading.
(a) The flight points at which dynamic gust analyses were conducted. The flight envelope is intended to match that of a typical modern commercial airliner, where, respectively,
${{V}_{\rm{C}}}$
and
${{\rm{M}}_{\rm{C}}}$
represent the calibrated airspeed and Mach in cruise. Similarly,
${{V}_{{\rm{MO}}}}$
,
${{\rm{M}}_{{\rm{MO}}}}$
and
${{Z}_{{\rm{MO}}}}$
are, respectively, the maximum operating calibrated airspeed, Mach and Altitude.
${{V}_{{\rm{stall}}}}$
is the calibrated airspeed at stall. (b) The gust frequencies investigated, plotted against
${{V}_\infty }$
. The frequency of the first wing bending mode is also shown.
The purpose of the passive gust load alleviation spoiler is to reduce internal wingbox loads. Therefore, the maximum and minimum envelope of wing root bending moment is computed across all five gust gradients at each flight point in order to quantify spoiler performance. The wing root is chosen because internal loads are greatest here, so a given load reduction offers the largest potential weight saving.
2.2 Passive spoiler implementation
This sub-section describes how the passive, strain-actuated spoiler developed and tested in our earlier work [Reference Wheatcroft, Shen, Groh, Pirrera and Schenk26–Reference Wheatcroft, Shen, Groh, Pirrera and Schenk28] is implemented in the aeroelastic model described above. The implementation requires computation of the wingbox strain at the given spoiler location and applying aerodynamic loads in response to spoiler deployment.
2.2.1 Strain calculation
For the purpose of computing the strain it experiences, the spoiler is assumed to be affixed to the primary wing structure, which is represented by a line of CBAR elements (Fig. 1(a)). Each element is modelled as an Euler-Bernoulli beam, with two orthogonal principal inertia axes shown as the
$2$
-axis and
$3$
-axis in Fig. 1(a). For each element, the weaker principal bending axis (the
$3$
-axis) is contained by the chord plane of the wing CAERO panels, and the element
$1$
-axis forms the centroidal axis. Deformations within each element can then be interpolated using the classic beam bending formula:
\begin{align}{w_i} = E{I_{jj}}\frac{{{d^4}{v_i}}}{{d{s^4}}}\end{align}
where
${w_i}$
and
${v_i}$
are, respectively, the loading per unit length and displacement along the
${i^{{\rm{th}}}}$
principal inertia axis of the element;
${I_{jj}}$
is the second moment of area about the
${j^{{\rm{th}}}}$
principal axis; and
$s$
is the arc length along the element
$1$
-axis (and
$i \ne j$
). NASTRAN CBARs prohibit the application of load within the length of the element, so Equation (1) can be solved with
${w_i} = 0$
for each wing CBAR element using the end node deformations as boundary conditions. This recovers the curvature,
${\kappa _{ii}} = {d^2}{v_j}/d{s^2}$
, about each principal axis, which can in turn be used to compute the total span-wise strain at a given cross-section and recovery point using (compressive strains positive):
\begin{align}{\varepsilon _{11}} = {d_2}{\kappa _{33}} + {d_3}{\kappa _{22}} + {\varepsilon _{{\rm{ax}}}},\end{align}
where
$\left[ {{d_2},{d_3}} \right]$
represents the coordinates of the recovery point within the cross-section in the principal axis system. The axial strain,
${\varepsilon _{{\rm{ax}}}}$
, is due to pure stretching of the element, which is computed directly from the displacements of the end nodes. The shear strain contribution due to torsion of the wingbox about the
$1$
-axis is neglected, as wing root torsional loads were found to be much smaller than root bending loads (for instance, the wing root torsion,
${M_1}$
, for the trim case plotted in Fig. 5 is an order of magnitude smaller than the wing root bending moment,
${M_3}$
). Furthermore, torsional stresses and strains depend not only on the wingbox depth, but also on the thickness of the wing skins and spars. This detailed design information is not captured by the simple CBAR elements used in the FFAST model.
For simplicity, the recovery point was set at
${d_3} = 0$
and
${d_2}$
equal to half the depth of the wingbox. This places the spoiler at the intersection of the strong bending axis and the upper surface of the wingbox, so strong axis bending has no effect on spoiler strain. In practice, this has a negligible effect, as
$\left| {{\kappa _{33}}\left| \gg \right|{\kappa _{22}}} \right|$
. The value of
${\varepsilon _{11}}$
at the spoiler recovery point is referred to as
${\varepsilon _{{\rm{spoiler}}}}$
and is calculated at each time step according to Equation (2).
It is assumed that spoiler deployment is driven purely by the longitudinal strain
${\varepsilon _{{\rm{spoiler}}}}$
. Clearly, the spoiler will be of finite length in the
$1$
-direction, and so will itself experience curvature along its length due to the relative rotation of its mounting points. However, this effect is neglected in the present work since the length of the spoiler along the
$1$
-axis is assumed to be small compared to the wing’s local radius of curvature.
In this work, we assume the convention that up-bend loading (lift) and the resulting internal moments are positive, and that compressive stresses and strains are positive. Spoiler performance is said to have improved when the spoiler reduces the range of loads that the wing must carry, or in other words when up-bend moments are decreased and when down-bend moments are increased (note that, in this context, the words ‘decreased’ and ‘increased’ are used in a strict mathematical sense).
2.2.2 Aerodynamic loads
The spoiler in our earlier work functions by locally reducing the wing’s
${c_l}$
[Reference Wheatcroft, Mahadik, Groh, Pirrera and Schenk27]. However, in the present work, the aerodynamic effect of the spoiler on the wing is modelled by applying downward loads to the structural mesh when the spoiler is deployed. This approach was chosen so NASTRAN’s in-built capability for modelling control surfaces could be used to compute the spoiler force, and is functionally equivalent to locally reducing lift.
The location of the starboard spoiler AESURF and hinge axis about which
${\rm{\delta }}$
is defined. The configuration is mirrored on the port wing.
The port and starboard spoilers are defined using AESURF cards and each consist of six aerodynamic panels as shown in Fig. 3. The chord of the spoiler is three panels, which matches the existing aileron control surface in the FFAST model. A span of two panels is used rather than one in order to give a more realistic aspect ratio. The spoiler was placed at the trailing edge (unlike the spoiler in our earlier work) as this gave the most realistic aerodynamic behaviour. The DMAP is used to extract the matrix mapping control surface deflections to forces on the mesh,
${Q_{{\rm{hx}}}}$
[Reference Castrichini, Siddaramaiah, Calderon, Cooper, Wilson and Lemmens5]. This is then used to obtain the force vector on the structural mesh caused by a unit spoiler control surface deployment angle
${{\hat{\bf f}}_{{\rm{spoiler}}}}$
. The total force vector from the spoiler is then given by
\begin{align}{{\bf{f}}_{{\rm{spoiler}}}} = \delta {\rm{\;}}{{\hat{\bf f}}_{{\rm{spoiler}}}},\end{align}
where
$\delta $
represents the spoiler control surface deployment angle about the axis shown in Fig. 3, with positive
$\delta $
resulting in upward spoiler deflections. Clearly, the linear relationship between
$\delta $
and
${{\bf{f}}_{{\rm{spoiler}}}}$
assumed by Equation (3) is an approximation of a much more complex reality. In particular, our earlier work showed that a lag often exists between spoiler deployment and a reduction in
${{\bf{f}}_{{\rm{spoiler}}}}$
[Reference Wheatcroft, Mahadik, Groh, Pirrera and Schenk27]. In Section 2.2.3, we introduce additional parameters (namely
${t_{{\rm{delay}}}}$
,
${t_{{\rm{dep}}}}$
,
${t_{{\rm{stow}}}}$
) to account for this effect.
The spoiler force vector
${{\bf{f}}_{{\rm{spoiler}}}}$
could be imposed on the structural mesh directly; however, it was found that the splining between the aerodynamic and structural meshes causes loads from the spoiler control surface to become spread over an unrealistically wide area of the structural mesh. So instead,
${{\bf{f}}_{{\rm{spoiler}}}}$
is concentrated into two point forces; one at either end of the CBAR in which the spoiler is located. These two forces vector are then scaled such that their total downward (global
$z$
-axis) component is the same as that of
${{\bf{f}}_{{\rm{spoiler}}}}$
. The two point forces act along the element
$2$
-axis, and their relative magnitude is such that their centre of pressure is at the spoiler span station of
$19$
m, or
$2/3$
of the wing semi-span. This is the location at which
${\varepsilon _{{\rm{spoiler}}}}$
is evaluated. This span station was selected as it gives a large lever arm relative to the wing root, but avoids the compliant wing tip region where the application of
${{\bf{f}}_{{\rm{spoiler}}}}$
causes excessive fluctuations in
${\varepsilon _{{\rm{spoiler}}}}$
. The choice of spoiler span-wise location is discussed further in Section 3.2.
No pitching moment was applied to the wing at the spoiler location. This is because, for the reasons discussed above, the spoiler control surface had to be placed at the trailing edge of the wing, so it does not accurately reflect the pitching moment contribution of the leading-edge tab design tested in our earlier work [Reference Wheatcroft, Mahadik, Groh, Pirrera and Schenk27]. In any case, spoiler induced pitching moments mainly impact torsional loads, which are not considered in this work as discussed above.
2.2.3 Deployment and stowage
Figure 4 schematically illustrates how the passive, strain-actuated behaviour of the spoiler was captured by linking the value of
${\varepsilon _{{\rm{spoiler}}}}$
to that of
$\delta $
. Accurately capturing the deployment behaviour of the physical spoiler prototype [Reference Wheatcroft, Mahadik, Groh, Pirrera and Schenk27] would require modelling its full nonlinear dynamics, as well as their coupling with the aerodynamics, which is beyond the scope of this work. Therefore, the spoiler behaviour was simplified based on the results of our earlier numerical and experimental work [Reference Wheatcroft, Shen, Groh, Pirrera and Schenk26, Reference Wheatcroft, Mahadik, Groh, Pirrera and Schenk27].
Schematic diagram showing the spoiler’s deployment and stowage behaviour in response to a hypothetical input spoiler strain,
${{\rm{\varepsilon }}_{{\rm{spoiler}}}}$
.
The spoiler remains stowed (
$\delta = 0$
) until the spoiler strain
${\varepsilon _{{\rm{spoiler}}}}$
exceeds the deployment strain
${\varepsilon _{{\rm{dep}}}}$
. This initiates the deployment sequence, which consists of first holding
$\delta $
at zero for a period
${t_{{\rm{delay}}}}$
, before deploying the spoiler by increasing
$\delta $
from zero to the full deployment angle
${\rm{\Delta }}$
at a constant rate over a time period
${t_{{\rm{dep}}}}$
. Once deployed, the spoiler remains so unless the stowage condition,
${\varepsilon _{{\rm{spoiler}}}} \lt {\varepsilon _{{\rm{stow}}}}$
is met, whereupon the stowage sequence is initiated. The stowage sequence operates in a similar way to the deployment sequence. Should the stowage condition be met at any time during the deployment sequence, then the stowage sequence is immediately started, and vice versa. Note that
${\varepsilon _{{\rm{stow}}}}$
is always less than
${\varepsilon _{{\rm{dep}}}}$
due to the inherent hysteresis in the spoiler’s structural response [Reference Wheatcroft, Mahadik, Groh, Pirrera and Schenk27, Reference Wheatcroft, Shen, Groh, Pirrera and Schenk28].
The lag time,
${t_{{\rm{delay}}}}$
, reflects the time taken for the spoiler to physically deploy. This depends on the nonlinear structural behaviour of the spoiler, and not on any aerodynamic effect [Reference Wheatcroft, Mahadik, Groh, Pirrera and Schenk27], so it does not vary throughout the flight envelope. We conservatively assume that the spoiler has no effect until
${t_{{\rm{delay}}}}$
has elapsed. Conversely,
${t_{{\rm{dep}}}}$
and
${t_{{\rm{stow}}}}$
, capture the time taken for the aerofoil’s lift coefficient to change in response to spoiler deployment. This aerodynamic lag effect is common to most spoiler designs [Reference Wheatcroft, Mahadik, Groh, Pirrera and Schenk27], and usually occurs over a fixed number of convective time units,
${T_c} = c/{V_\infty }$
, where
$c$
is the wing chord and
${V_\infty }$
is the true airspeed of the freestream. We therefore define
${t_{{\rm{dep}}}}$
and
${t_{{\rm{stow}}}}$
in units of
${T_c}$
(with
$c = 4.53$
m at the spoiler location), so their absolute value in seconds varies throughout the flight envelope.
The deployment parameters
${\varepsilon _{{\rm{dep}}}}$
,
${\varepsilon _{{\rm{stow}}}}$
,
${t_{{\rm{delay}}}}$
,
${t_{{\rm{dep}}}}$
,
${t_{{\rm{stow}}}}$
and
${\rm{\Delta }}$
are treated as design variables of the spoiler, and investigating their effect on spoiler performance is one objective of the present work.
The behaviour shown in Fig. 4 is designed to capture the nonlinear relationship between
${\varepsilon _{{\rm{spoiler}}}}$
and
$\delta $
exhibited by the spoiler in our earlier work [Reference Wheatcroft, Shen, Groh, Pirrera and Schenk26–Reference Wheatcroft, Shen, Groh, Pirrera and Schenk28]. This relationship is the key to defining the influence of the spoiler on the aeroelastic response of the overall airframe. So, despite the fact that the spoiler model is greatly simplified in the present work, this key aspect of its nonlinear mechanics is still captured. However, clearly this simplified model cannot capture all aspects of the spoiler structure’s complex nonlinear mechanics, and implementing a higher fidelity spoiler model would be an important area for future work.
2.3 Model capabilities
The FFAST model represents the aircraft wing structure using beam elements, which accurately models the stiffness of the aircraft, but is limited in its ability to resolve local strains, especially with a comparatively coarse structural mesh. The fidelity of the model is nonetheless sufficient for the analysis conducted herein, as this work is concerned with relative changes in strain and moment at particular wing locations, rather than their absolute values. However, there are naturally some limitations to using a lower fidelity model. Highly localised effects are not well captured, particularly effects that depend on load-paths within the wingbox. One such effect is the influence of
${{\bf{f}}_{{\rm{spoiler}}}}$
on strains in the immediate vicinity of the spoiler, which naturally impacts the value of
${\varepsilon _{{\rm{spoiler}}}}$
. Intuitively, the lift reduction caused by the spoiler control surface will have a direct impact on the strain the spoiler structure experiences. However, the strength of this effect depends heavily on the precise load path of external aerodynamic pressures into the primary wingbox structure, which the simplified structural model cannot capture. Furthermore, by introducing spoiler loads at the ends of a given element, the spoiler load is spread over an unrealistically large portion of the wing, magnifying the lever arm of spoiler forces relative to the location at which
${\varepsilon _{{\rm{spoiler}}}}$
is computed. Thus, spreading out the spoiler loads causes an unrealistically high degree of coupling between
${{\bf{f}}_{{\rm{spoiler}}}}$
and
${\varepsilon _{{\rm{spoiler}}}}$
. This has particular ramifications when assessing the possibility of the spoiler inducing limit cycle oscillations in the wing, a detailed treatment of which is given in Section 3.2.3.
3.0 Results and discussion
Analysis results are split into two sections. First, Section 3.1 presents results from the static trim analyses of the model, which determines the minimum value of
${\varepsilon _{{\rm{dep}}}}$
and
${\varepsilon _{{\rm{stow}}}}$
to prevent premature spoiler deployment and ensure correct stowage. Next, Section 3.2 presents results from the dynamic gust analyses which are used to investigate the load alleviation performance of the spoiler.
3.1 Steady state results
3.1.1 Span-wise loads distribution
Figure 5 shows bending moment
${M_3}$
, shear force
${S_{12}}$
, curvature
${\kappa _{33}}$
and wing skin span-wise strain
${\varepsilon _{11}}$
plotted against span station under steady state trim at
$Z = 7,500$
m,
$M = 0.89$
. This is the flight point at which wing root bending moment is greatest. Blue and pink lines are the distributions with the spoiler stowed and deployed, respectively. As expected, the bending moment varies linearly and shear is constant within each element. Curvature is discontinuous at element boundaries because second moment of area
${I_{33}}$
varies discretely from element to element, resulting in the sawtooth curvature pattern. Strain is therefore also discontinuous at the element boundaries since it is directly proportional to curvature. As discussed in Section 2.3, this discontinuity is not of great concern for our analysis because the spoiler behaviour relies on changes in
${\varepsilon _{{\rm{spoiler}}}}$
, and not its absolute value. However, care should be taken when comparing strains between different span stations. The pink lines on Fig. 5 show the same data with the spoiler deployed to
${15^ \circ }$
at a span station of
$19$
m. Wing root bending moment
${M_3}$
is reduced due to the fact that the centre of lift has moved closer to the wing root. The moment and shear force outboard of the spoiler differ slightly between the stowed and deployed cases because the trim angle-of-attack is slightly increased when the spoiler is deployed.
Steady state trim distributions of (a) bending moment
${{M}_3}$
, (b) shear force
${{S}_{12}}$
, (c) curvature
${{\rm{\kappa }}_{33}}$
and (d) wing skin span-wise strain
${{\rm{\varepsilon }}_{11}}$
at
${{Z}} = 7, \! 500$
m and
${\rm{M}} = 0.89$
.
${{\rm{\varepsilon }}_{11}}$
is computed at a recovery point of
${{d}_1} = 0$
and
${{d}_2}$
equal to half the wingbox depth.
3.1.2 Flight envelope-wide loads distribution
Figure 6 shows
${\varepsilon _{{\rm{spoiler}}}}$
in the
$1$
g trim configuration with the spoiler stowed throughout the flight envelope (i.e. the value of the blue line on Fig. 5(d) for span
$ = 19$
m at every flight point). The purpose of this plot is to establish the maximum
$1$
g trim value of
${\varepsilon _{{\rm{spoiler}}}}$
across all flight points, which is
\begin{align*}{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}} = 434{\rm{\;}}\mu \varepsilon .\end{align*}
This is an important design parameter for the spoiler, as it sets the minimum threshold for both
${\varepsilon _{{\rm{dep}}}}$
and
${\varepsilon _{{\rm{stow}}}}$
. The deployment strain must be greater than
${\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
in order to prevent spoiler deployment during
$1$
g flight. Further, if the stowage strain is smaller than
${\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
then the spoiler will not always stow once the gust has passed and strain returns to its steady state value. In practice, it is desirable to make
${\varepsilon _{{\rm{dep}}}}$
somewhat larger than
${\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
so as to prevent the spoiler from deploying during day-to-day aircraft manoeuvres. CS-25 also mandates that aircraft be able to perform a collision-avoidance manoeuvre with a load factor of up to
$2.5$
g, depending on the flight point [8]. It is likely that spoiler deployment during this manoeuvre would be undesirable, as it would lead to a sudden change in the handling characteristics of the aircraft. The implications of this are discussed further in Section 3.2.3.
3.2 Transient results
This section presents the time-varying results obtained during a 1MC gust encounter. First, we consider the baseline case without passive spoiler deployment, before studying the spoiler’s load alleviation performance. In all cases,
${M_i}$
refers to the internal moment stress resultant within the wing, i.e.
$E{I_{ii}}{\kappa _{ii}}$
.
The strain at the spoiler location,
${{\rm{\varepsilon }}_{{\rm{spoiler}}}}$
, under
$1$
g trim with the spoiler stowed at the
$16$
flight points considered (areas between flight points interpolated linearly). The maximum value,
${{\rm{\varepsilon }}_{{\rm{spoiler}},1{\rm{g}}}} = 434$
${\rm{\mu \varepsilon }}$
, provides a lower limit on the value of
${{\rm{\varepsilon }}_{{\rm{dep}}}}$
and
${{\rm{\varepsilon }}_{{\rm{stow}}}}$
.
3.2.1 Without passive spoiler
Here, the transient loads in the absence of the passive spoiler are used to establish a baseline case against which spoiler performance can be benchmarked.
Baseline Loads
Figure 7 shows bending moment
${M_3}$
at the wing root (denoted
${M_{3,{\rm{root}}}}$
) vs time during a gust encounter without passive spoiler deployment at
$Z = 7,500$
m,
${\rm{M}} = 0.89$
and
$H = 76.2$
m. This is the flight point and gust length which causes the largest wing root moment, so it is referred to as the ‘sizing case’. As expected, the gust results in an initial increase in
${M_{3,{\rm{root}}}}$
, followed by a decaying transient. This transient is modulated onto a much lower frequency oscillation, which rises from
$t \approx 1$
s until
$t \approx 3$
s, overshooting the initial steady state
${M_{3,{\rm{root}}}}$
before fully decaying. This low frequency oscillation is caused by the aircraft’s changing angle-of-attack
${\alpha _{{\rm{tot}}}}$
during the simulationFootnote 1. The secondary
$y$
-axis in Fig. 7 shows
${\alpha _{{\rm{tot}}}}$
, which decreases during the initial phase of the simulation, before undergoing a slightly under-damped decay due to the inherent longitudinal stability of the aircraft. In reality, the aircraft’s flight control system would likely use the elevator to damp out these oscillations in
${\alpha _{{\rm{tot}}}}$
much faster.
Wing root bending moment
${{M}_3}$
and total angle-of-attack
${{\rm{\alpha }}_{{\rm{tot}}}}$
vs time without the passive spoiler during the sizing gust encounter (
${{Z}} = 7, \! 500$
m,
${\rm{M}} = 0.89$
and
${{H}} = 76.2$
m).
Figure 7 pertains to a single flight point and gust length; however, an aircraft must be designed to withstand loads experienced across the whole flight envelope. Therefore, a baseline envelope of loads across the flight envelope is constructed by taking the maximum and minimum values of
${M_{3,{\rm{root}}}}$
across all simulation times and gust lengths at each flight point. These maxima and minima are plotted in Fig. 8, and provide a benchmark against which the spoiler performance can be compared. We refer to moments lying between these maximum and minimum values as being ‘within’ the baseline envelope. This terminology should not be confused with the flight envelope shown in Fig. 2(a). As stated above, the flight point with the largest
${\rm{Max}}\left( {{M_{3,{\rm{root}}}}} \right)$
and smallest
${\rm{Min}}\left( {{M_{3,{\rm{root}}}}} \right)$
is
$Z = 7, \! 500$
m,
${\rm{M}} = 0.89$
. An overall sizing value of
${M_{3,{\rm{root}}}}$
for the baseline aircraft is established according to
\begin{align}{{M_{{\rm{B}},{\rm{sizing}}}}} {} &= {\rm{Max}}\left[ {\left| {{\rm{Max}}\left( {{M_{3,{\rm{root}}}}} \right)\left| , \right|{\rm{Min}}\left( {{M_{3,{\rm{root}}}}} \right)} \right|} \right]\nonumber \\
&= {\rm{Max}}\left[ {\left| {16.3{\rm{\;MNm}}\left| , \right| \! - \! 1.26{\rm{\;MNm}}} \right|} \right] = 16.3{\rm{\;MNm}}.\end{align}
This is the root bending moment to which the baseline wing must be designed, and which the spoiler is designed to reduce. Note that Equation (4) implicitly assumes that up-bend and down-bend moments are equally damaging to the wing. This is not true in general since a wingbox cross section is not typically symmetric about the
$3$
-axis, and because structures in compression must be designed against bucking as well as material failure. However, in this case the maximum up-bend moment is much greater than the minimum down-bend moment, so it is assumed that designing to
${M_{{\rm{B}},{\rm{sizing}}}}$
would produce a wing strong enough to also carry the worst down-bend loads. Furthermore, the worst down-bend moments are more likely to be encountered during a hard landing than during a 1MC gust, so the true sizing down-bend moment is likely to be lower than Fig. 8(b) suggests.
The (a) maximum and (b) minimum value of
${{M}_{3,{\rm{root}}}}$
across all simulation times and gust lengths at each flight point in the baseline case with no passive spoiler (areas between flight points interpolated linearly). The largest overall magnitude of
${{M}_{3,{\rm{root}}}}$
provides the sizing load to which the wing must be designed. (c) The maximum and minimum
${{M}_{3,{\rm{root}}}}$
visualised on the same set of axes. Values of
${{M}_{3,{\rm{root}}}}$
, which are between the blue (maximum) and pink (minimum) surfaces are said to lie within the baseline envelope.
Span-wise Spoiler Placement
Next, we briefly consider the selection of the span-wise location for a passive spoiler, based on the results from the baseline model. All other things being equal, a spoiler should provide a greater reduction in
${M_{3,{\rm{root}}}}$
if it is placed nearer the wing tip, since this maximises its lever arm relative to the wing root. However, this assumes that spoiler deployment is triggered at the same time regardless of its span station; a spoiler which deploys after the peak
${M_{3,{\rm{root}}}}$
has occurred will clearly provide no load alleviation benefit. The instant of spoiler deployment at a given span station depends on when span-wise strain
${\varepsilon _{11}}$
exceeds the chosen deployment value
${\varepsilon _{{\rm{dep}}}}$
during a gust encounter. Span-wise strain is driven by
${M_3}$
, so here we assess when a spoiler might deploy at each span station by considering when the peak
${M_3}$
occurs throughout the wing during a gust encounter.
Figure 9(a) shows
${M_3}$
at every span station and time during the analysis at the sizing flight point and gust gradient. Note that Fig. 7 is the intersection of Fig. 9(a) with the span station
$ = 0$
plane. The locus of highest bending moment at each span is shown as the pink line. This locus is also shown against time in Fig. 9(b), where the time of maximum moment at the wing root and wing tip are denoted
${t_{{\rm{M}},{\rm{root}}}}$
and
${t_{{\rm{M}},{\rm{tip}}}}$
, respectively. Also shown is the time at which the peak of the 1MC gust reaches a given span. As expected, the sweep of the wing means that the gust peak arrives at the wing tip (
${t_{{\rm{g}},{\rm{tip}}}}$
) after it arrives at the root (
${t_{{\rm{g}},{\rm{root}}}}$
). However, Fig. 9(b) shows that peak moment occurs at the tip before it occurs at the root (i.e.
${t_{{\rm{M}},{\rm{root}}}} \gt {t_{{\rm{M}},{\rm{tip}}}}$
), despite the fact that the gust peak arrives at the tip after it arrives at the root. This behaviour is useful, because it means that a spoiler placed near the wing tip not only has a larger lever arm over the wing root, but also has ‘advance warning’ of the worst wing root bending moments, which allows for an earlier deployment. Early deployment is shown to provide better load alleviation performance in Section 3.2.2.
(a)
${{M}_3}$
plotted against span station and simulation time; the locus of maximum
${{M}_3}$
for a given span station is shown in pink. (b) The locus of maximum
${{M}_3}$
plotted against time, along with the arrival time of the gust peak. The peak moment occurs at the wing tip before it occurs at the root. (c) The tip-to-root delay in peak bending moment,
${{t}_{{M},{\rm{tip}}}} - {{t}_{{M},{\rm{root}}}}$
, for all flight points and gust lengths, and delay in peak gust loading,
${{t}_{{\rm{g}},{\rm{tip}}}} - {{t}_{{\rm{g}},{\rm{root}}}}$
, plotted against true airspeed. The delay in moment is less than the delay in gust arrival in all cases, showing that the wing tip receives ‘advance warning’ of the gust.
In practice, there may be a trade-off with other factors when determining the span-wise location of the spoiler. Firstly, as discussed above, the wing is more compliant near the tip so the application of
${{\bf{f}}_{{\rm{spoiler}}}}$
may cause excessive fluctuations in
${\varepsilon _{{\rm{spoiler}}}}$
(see Section 3.2.3). Secondly, the chord of most wings is tapered with span, meaning the wing tip generates much less lift per span,
${c_l}$
, than the root. Thus, a spoiler of a given size which functions by reducing
${c_l}$
will provide less absolute lift reduction if placed nearer the tip.
The delay between peak moment at the wing tip and wing root,
${t_{{\rm{M}},{\rm{tip}}}} - {t_{{\rm{M}},{\rm{root}}}}$
, is plotted against true airspeed for all flight points in Fig. 9(c). Also shown is the delay between the gust peak arriving at the root and the tip,
${t_{{\rm{g}},{\rm{tip}}}} - {t_{{\rm{g}},{\rm{root}}}}$
. This plot shows that the peak moment occurs at the tip before the root for the majority of flight points, and that any lag is always less than the delay caused by the later arrival of the gust peak at the wing tip. So the conclusions stated above apply throughout the flight envelope, and not just at the sizing case. This result suggests that, in most cases, peak wing bending deformations are dominated by the first wing bending vibrational mode. So deformations at each span station remain largely in phase with one another during the early part of the gust when peak deformations occur.
3.2.2 With passive spoiler
In this sub-section we present results of gust encounters with the passive spoiler activated. Initially, results for a single flight point and set of spoiler parameters are presented to demonstrate the spoiler’s basic behaviour. Next, the impact of changing spoiler design parameters across the whole flight envelope is investigated.
Single Flight Point Analysis
Figure 10 shows spoiler strain
${\varepsilon _{{\rm{spoiler}}}}$
, wing root bending moment
${M_{3,{\rm{root}}}}$
and spoiler deployment
$\delta $
all plotted against time during the sizing gust encounter with passive spoiler deployment. The
$y$
-axis of the moment plot has been normalised as
\begin{align}{\hat M_{3,{\rm{root}}}} = \frac{{{M_{3,{\rm{root}}}}}}{{{M_{{\rm{B}},{\rm{sizing}}}}}},\end{align}
so that an
${\hat M_{3,{\rm{root}}}}$
of less than
$1$
means that the spoiler is providing a load alleviation benefit to the wing. The baseline
${\varepsilon _{{\rm{spoiler}}}}$
and
${\hat M_{3,{\rm{root}}}}$
are also shown for comparison. The spoiler parameters are
${\varepsilon _{{\rm{dep}}}} = 1.15{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
,
${\varepsilon _{{\rm{stow}}}} = 1.1{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
,
${t_{{\rm{delay}}}} = 0$
s,
${t_{{\rm{dep}}}} = {t_{{\rm{stow}}}} = 2{T_c}$
and
${\rm{\Delta }} = 15$
${{\rm}^{ \! \circ} }$
. The sensitivity of the spoiler’s performance to these parameters is investigated below. This particular configuration is selected to provide a clear demonstration of the spoiler’s typical behaviour during a single gust encounter. In practice, there will be physical bounds on each of the spoiler parameters, which depend on the specific design of the spoiler and that of the wider airframe.
(a)
${{\rm{\varepsilon }}_{{\rm{spoiler}}}}$
vs time during the sizing gust encounter, with passive spoiler deployment shown on the secondary
${{y}}$
-axis. Baseline
${{\rm{\varepsilon }}_{{\rm{spoiler}}}}$
is also shown. (b)
${{\hat{ M}}_{3,{\rm{root}}}}$
vs time for the same gust encounter. Passive spoiler activation has reduced
${\rm{Max}}\left( {{{M}_{3,{\rm{root}}}}} \right)$
by around
$17$
%.
The plot of
${\varepsilon _{{\rm{spoiler}}}}$
(Fig. 10(a)) demonstrates that the spoiler functions as intended, with deployment and stowage occurring in response to
${\varepsilon _{{\rm{spoiler}}}}$
traversing
${\varepsilon _{{\rm{dep}}}}$
and
${\varepsilon _{{\rm{stow}}}}$
as expected. The plot of
${\hat M_{3,{\rm{root}}}}$
(Fig. 10(b)) shows that the spoiler provides a load alleviation benefit to up-bend moments of around
$17$
% at this particular flight point and gust length. The spoiler has also improved down-bend performance by increasing the minium value of
${\hat M_{3,{\rm{root}}}}$
. Initially, this seems counter-intuitive, since
${{\bf{f}}_{{\rm{spoiler}}}}$
acts downward, so might be expected to worsen down-bend moments. However, the downward action of
${{\bf{f}}_{{\rm{spoiler}}}}$
also has the effect of reducing the elastic energy stored in the wing during the up-bend phase of the gust, so there is less overshoot of the steady state
${M_{3,{\rm{root}}}}$
in the down-bend phase. In addition,
${\varepsilon _{{\rm{spoiler}}}}$
is slightly reduced while the spoiler is deployed, which brings forward the instant that
${\varepsilon _{{\rm{spoiler}}}}$
falls back below
${\varepsilon _{{\rm{stow}}}}$
. The earlier spoiler stowage reduces the portion of the down-bend phase for which the spoiler is deployed, further improving down-bend performance. These two effects do not always fully counteract the inherent negative impact of
${{\bf{f}}_{{\rm{spoiler}}}}$
on down-bend moments, as is demonstrated later.
Sensitivity to
${\varepsilon _{{\bf{dep}}}}$
&
${\varepsilon _{{\bf{stow}}}}$
We now explore how the spoiler’s load alleviation performance across the flight envelope is affected by key spoiler parameters, starting with the deployment and stowage strains,
${\varepsilon _{{\rm{dep}}}}$
and
${\varepsilon _{{\rm{stow}}}}$
. Initially,
${\varepsilon _{{\rm{dep}}}}/{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
was varied from
$1.15$
to
$1.9$
while maintaining the relation
${\varepsilon _{{\rm{stow}}}} = {\varepsilon _{{\rm{dep}}}} - 0.05{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
and holding all other spoiler parameters constant at the values given in the previous section. The deployment and stowage strains were varied together in order to maintain a constant amount of hysteresis in the spoiler response. The maximum and minimum values of
${{\hat{ M}}_{3,{\rm{root}}}}$
(i.e. worst up-bend and down-bend moments) were then computed at every flight point, just as they were for the baseline case in Fig. 8.
Plots depicting the sensitivity of
$\,{{\hat{M}}_{3,{\rm{root}}}}$
to
${{\rm{\varepsilon }}_{{\rm{dep}}}}/{{\rm{\varepsilon }}_{{\rm{spoiler}},1{\rm{g}}}}$
at all
$16$
flight points. Each sub-plot represents a different flight point, with each one showing the maximum (upper line) and minimum (lower line) enveloping value of
$\,{{\hat{M}}_{3,{\rm{root}}}}$
across all gust lengths for six different values of
${{\rm{\varepsilon }}_{{\rm{dep}}}}/{{\rm{\varepsilon }}_{{\rm{spoiler}},1{\rm{g}}}}$
. The value of
$\,{{\hat{M}}_{3,{\rm{root}}}}$
is also mapped to the colour scale shown on the right to aid comparison between sub-plots. Enveloping values of
${{\hat{M}}_{3,{\rm{root}}}}$
that are inside and outside the baseline envelope are denoted by black crosses and dots, respectively. Sizing cases for each of the six different values of
${{\rm{\varepsilon }}_{{\rm{dep}}}}$
are additionally denoted by a hollow circle. The sub-plots are arranged in the same grid-like order as the flight points in Fig. 2(a).
The results of these analyses are summarised in Fig. 11, in which each sub-plot shows the sensitivity of
${\hat M_{3,{\rm{root}}}}$
to
${\varepsilon _{{\rm{dep}}}}/{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
at a given flight point. The
$16$
sub-plots are arranged in the same grid-like order as the
$16$
flight points in Fig. 2(a). Each sub-plot shows maximum (upper line) and minimum (lower line) enveloping values of
${\hat M_{3,{\rm{root}}}}$
across all gust lengths on its
$y$
-axis, and the corresponding value of
${\varepsilon _{{\rm{dep}}}}/{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
on its
$x$
-axis. Thus, each sub-plot shows the sensitivity of enveloping
${\hat M_{3,{\rm{root}}}}$
to
${\varepsilon _{{\rm{dep}}}}/{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
at a particular flight point. Note that, in order to make this sensitivity as clear as possible, each
$y$
-axis is split and their ranges are not consistent among the sub-plots. Thus, the colour of each line is additionally mapped to the value of
${\hat M_{3,{\rm{root}}}}$
to aid comparison between sub-plots, as shown by the colour bar on the right.
A black cross (
$ \times $
) denotes an
${\hat M_{3,{\rm{root}}}}$
which lies within the baseline envelope (Fig. 8) at a given flight point. An
${\hat M_{3,{\rm{root}}}}$
lying on or outside the baseline envelope is denoted by a black dot (
$ \bullet $
). Areas shaded grey are outside the baseline envelope. The boundaries of the baseline envelope vary between flight points, as shown in Fig. 8.
Any point on Fig. 11 which is the sizing maximum or minimum
${\hat M_{3,{\rm{root}}}}$
for a given value of
${\varepsilon _{{\rm{dep}}}}/{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
is additionally denoted by a hollow circle (
$ \circ $
). These values are amalgamated and plotted against
${\varepsilon _{{\rm{dep}}}}/{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
in Fig. 12(a), showing how the sizing up-bend and down-bend loads are influenced by
${\varepsilon _{{\rm{dep}}}}$
.
Figure 11 shows that, at most flight points, reducing
${\varepsilon _{{\rm{dep}}}}$
improves the up-bend load alleviation performance of the spoiler, i.e. the maximum wing root bending moment is reduced. This result is corroborated by intuition; the spoiler can deploy earlier in the gust when
${\varepsilon _{{\rm{dep}}}}$
is smaller. As
${\varepsilon _{{\rm{dep}}}}$
is increased, there comes a point where the gust induced wingbox strain no longer exceeds
${\varepsilon _{{\rm{dep}}}}$
and the spoiler does not deploy. This is the case for almost all
${\varepsilon _{{\rm{dep}}}}$
at lower Mach, because the peak
${\varepsilon _{{\rm{spoiler}}}}$
achieved during a gust is much lower in this part of the flight envelope (c.f. Fig. 8(a)).
(a) Sensitivity of the sizing values of
${{\hat{M}}_{3,{\rm{root}}}}$
in Fig. 11 to
${{\rm{\varepsilon }}_{{\rm{dep}}}}$
. (b) Sensitivity of the sizing
${{\hat{M}}_{3,{\rm{root}}}}$
to
${{\rm{\varepsilon }}_{{\rm{dep}}}}$
with
${{\rm{\varepsilon }}_{{\rm{stow}}}}$
held constant at
$1.1{{\rm{\varepsilon }}_{{\rm{spoiler}},1{\rm{g}}}}$
. Up-bend performance is unchanged, however most sizing down-bend moments are now slightly outside the baseline envelope.
The effect of the spoiler on down-bend moments is similar to its effect on up-bend moments; for each flight point there is an
${\varepsilon _{{\rm{dep}}}}$
above which the spoiler does not deploy, and thus has no effect on loads. However, there are a number of flight points where the spoiler has worsened down-bend performance, and
${\rm{Min}}( {{{\hat M}_{3,{\rm{root}}}}} )$
lies outside the baseline envelope. This is only of potential concern if such a point is also the sizing value, as is the case at
$Z = 7,500$
m and
$M = 0.89$
for
${\varepsilon _{{\rm{dep}}}}/{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}} = 1.15$
. However, in this case, the sizing down-bend
${\hat M_{3,{\rm{root}}}}$
is only marginally less than the baseline value, and is therefore unlikely to govern the sizing of the wing.
The sizing values of
${\hat M_{3,{\rm{root}}}}$
across all 16 flight points are amalgamated and plotted against
${\varepsilon _{{\rm{dep}}}}/{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
in Fig. 12(a), as stated above. The sizing loads mirror the trends at each individual flight point: decreasing
${\varepsilon _{{\rm{dep}}}}$
and
${\varepsilon _{{\rm{stow}}}}$
means the spoiler is deployed for longer, thereby reducing up-bend moments, but also increasing down-bend moments at certain values of
${\varepsilon _{{\rm{dep}}}}$
. Figure 12(a) also shows that the spoiler performs well in absolute terms across a range of
${\varepsilon _{{\rm{dep}}}}$
, providing a load alleviation benefit of at least
$10$
% for
$1.15 \le {\varepsilon _{{\rm{dep}}}}/{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}} \le 1.45$
.
Figure 12(b) shows the sensitivity of
${\hat M_{3,{\rm{root}}}}$
to the same range of
${\varepsilon _{{\rm{dep}}}}$
as in Fig. 12(a), but now
${\varepsilon _{{\rm{stow}}}}$
is held constant at
$1.1{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
. This change has no effect on deployment and thus up-bend performance, but it does delay spoiler stowage until later in the gust. In most cases this means that
${{\bf{f}}_{{\rm{spoiler}}}}$
is acting on the wing well into the down-bend phase, worsening the down-bend performance of the spoiler. However, in all cases the sizing down-bend moment remains small compared to the corresponding up-bend moment, and so the spoiler can still be said to provide a load alleviation benefit of at least
$10$
% for
$1.15 \le {\varepsilon _{{\rm{dep}}}}/{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}} \le 1.45$
.
The results presented in Fig. 12 suggest that the load alleviation benefit of the spoiler is maximised by setting the deployment strain,
${\varepsilon _{{\rm{dep}}}}$
, as close to the
$1$
g value as possible. As discussed in Section 3.1, the aircraft must be designed to withstand a high load factor collision-avoidance manoeuvre, during which spoiler deployment is unlikely to be permissible. A conservative estimate of
${\varepsilon _{{\rm{spoiler}}}}/{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
during this manoeuvre is
$2.5$
[8], but Fig. 12 shows that a spoiler with an
${\varepsilon _{{\rm{dep}}}}/{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
of
$1.75$
or more can provide no load alleviation benefit. This suggests that the spoiler should be prevented from deploying during the collision-avoidance manoeuvre, or that the aircraft must be capable of executing the manoeuvre with the spoiler deployed.
Sensitivity to
${\bf{\Delta }}$
Next, the effect of the maximum spoiler deployment angle,
${\rm{\Delta }}$
, is investigated, as this sets the maximum magnitude of
${{\bf{f}}_{{\rm{spoiler}}}}$
(Equation (3)). Six values of
${\rm{\Delta }}$
are considered, ranging from
${2^ \circ }$
to
${16^ \circ }$
, with sizing moments shown in Fig. 13. Increasing
${\rm{\Delta }}$
linearly reduces sizing up-bend moments. There is little effect from
${\rm{\Delta }}$
on sizing down-bend moments, other than a slight reduction until
${\rm{\Delta }} = {12^ \circ }$
followed by a slight increase. This increase is due to the effect mentioned above, where the value of
${\varepsilon _{{\rm{spoiler}}}}$
is reduced when the spoiler is deployed. These results show, as expected, that the spoiler should provide as large a lift reduction as possible upon deployment in order to maximise its load alleviation performance.
Plot showing the sensitivity of the sizing
${{\hat{M}}_{3,{\rm{root}}}}$
to
${\rm{\Delta }}$
. As expected, increasing
${\rm{\Delta }}$
reduces the sizing up-bend moment. Increasing
${\rm{\Delta }}$
also slightly decreases the sizing down-bend moment, but this remains small compared to the up-bend value.
Sensitivity to
${t_{{\bf{delay}}}}$
and
${t_{{\bf{dep}}}}$
We now investigate the sensitivity of the spoiler’s performance to the two time periods
${t_{{\rm{delay}}}}$
and
${t_{{\rm{dep}}}}$
. Initially, we focus on
${t_{{\rm{delay}}}}$
, which is the length of time between the deployment condition
${\varepsilon _{{\rm{spoiler}}}} \gt {\varepsilon _{{\rm{dep}}}}$
being met and lift reduction starting. Similarly, lift does not begin to increase until
${t_{{\rm{delay}}}}$
after the stowage condition
${\varepsilon _{{\rm{spoiler}}}} \lt {\varepsilon _{{\rm{stow}}}}$
has been met (c.f. Fig. 4).
The relationship between
${t_{{\rm{delay}}}}$
and the sizing moments is shown in Fig. 14(a). The plot shows that the up-bend and down-bend performance of the spoiler is improved when
${t_{{\rm{delay}}}}$
is reduced. The sizing down-bend moments are all outside of the baseline envelope, although for the most part remain significantly smaller than the sizing up-bend moment. However, at
${t_{{\rm{delay}}}} \geqslant 0.1$
s, the sizing down-bend moment is roughly triple the baseline value, and its magnitude is nearly a quarter of
${M_{{\rm{B}},{\rm{sizing}}}}$
. In a real wingbox, this significantly decreased down-bend moment could necessitate reinforcement of the bottom skin against buckling, partially negating any weight saving resulting from the reduced up-bend moment. In summary, the results in Fig. 14(a) show that the passive spoiler should be designed to minimise
${t_{{\rm{delay}}}}$
. In practice, this is achieved by reducing the inertia of its moving parts, and increasing the amount of elastic energy released by the morphing structure during deployment [Reference Wheatcroft, Mahadik, Groh, Pirrera and Schenk27].
Finally, the sensitivity of the spoiler performance to
${t_{{\rm{dep}}}}$
is investigated. Recall that
${t_{{\rm{dep}}}}$
is the time taken for
$\delta $
to transition from
$0$
to
${\rm{\Delta }}$
, and vice-versa for
${t_{{\rm{stow}}}}$
. For the purposes of this study, the relationship
${t_{{\rm{dep}}}} = {t_{{\rm{stow}}}}$
is maintained since both of these times reflect the aerodynamic lag introduced by the spoiler, so any change in
${t_{{\rm{dep}}}}$
is likely to be matched by a change in
${t_{{\rm{stow}}}}$
. Figure 14(b) shows the relationship between the sizing moments and
${t_{{\rm{dep}}}}$
. Performance of the spoiler in both up-bend and down-bend is improved by reducing
${t_{{\rm{dep}}}}$
and
${t_{{\rm{stow}}}}$
. However, the spoiler still performs well even at the larger deployment times considered, with
${t_{{\rm{dep}}}} = {t_{{\rm{stow}}}} = 10{T_c}$
providing a
$10$
% reduction in up-bend moment. This is noteworthy because
$10{T_c}$
corresponds to
$164$
ms at the sizing flight point, and yet spoiler performance is better in this case than when
${t_{{\rm{delay}}}}$
is only
$75$
ms. This is because
${t_{{\rm{delay}}}}$
corresponds to a period where
${{\bf{f}}_{{\rm{spoiler}}}} = 0$
, whilst
${{\bf{f}}_{{\rm{spoiler}}}}$
is constantly increasing during
${t_{{\rm{dep}}}}$
. Thus, Fig. 14(b) shows that a short deployment and stowage time is desirable, but also that rapidly applying any amount of
${{\bf{f}}_{{\rm{spoiler}}}}$
is beneficial, even if achieving the final value takes longer.
(a) Plot showing the sensitivity of the sizing
${{\hat{M}}_{3,{\rm{root}}}}$
to
${{t}_{{\rm{delay}}}}$
. Reducing
${{t}_{{\rm{delay}}}}$
improves spoiler performance in both up-bend and down-bend. (b) The sensitivity of the sizing
${{\hat{M}}_{3,{\rm{root}}}}$
to
${{t}_{{\rm{dep}}}}$
(with
${{t}_{{\rm{stow}}}} = {{t}_{{\rm{dep}}}}$
). Again, reducing
${{t}_{{\rm{dep}}}}$
and
${{t}_{{\rm{stow}}}}$
improves spoiler performance in both up-bend and down-bend, however the effect is not so strong as it is for
${{t}_{{\rm{delay}}}}$
.
It should be noted that, for simplicity,
${t_{{\rm{delay}}}}$
and
${t_{{\rm{dep}}}}$
have here been treated as independent parameters of the spoiler. The former is intended to model the time taken for the spoiler to physically deploy, while the latter is intended to model the time taken for the separated airflow to develop around the deployed spoiler. Clearly there is overlap between these time periods in reality; flow is altered as soon as the spoiler starts to deploy, and this can immediately influence the lift force generated by the wing [Reference Wheatcroft, Mahadik, Groh, Pirrera and Schenk27]. Thus, a more complete knowledge of the spoiler’s transient aerodynamic behaviour is necessary in order to improve the modelling fidelity and thus accuracy of these results.
3.2.3 Spoiler induced oscillations
Lastly, we investigate the possibility that repeated deployment and stowage of the spoiler could cause the wing to enter a limit cycle oscillation. This is a situation in which the spoiler deploys in response to
${\varepsilon _{{\rm{spoiler}}}}$
exceeding
${\varepsilon _{{\rm{dep}}}}$
, but then stows due to the reduction in
${\varepsilon _{{\rm{spoiler}}}}$
caused by the downward action of
${{\bf{f}}_{{\rm{spoiler}}}}$
. With the spoiler stowed,
${\varepsilon _{{\rm{spoiler}}}}$
could once again increase above
${\varepsilon _{{\rm{dep}}}}$
and the cycle would repeat.
An example of this process is shown in Fig. 15(a), which shows
${\varepsilon _{{\rm{spoiler}}}}$
and
$\delta $
plotted against time for
$Z = 13,100$
m,
${\rm{M}} = 0.89$
and
$H = 76.2$
m. The spoiler parameters are
${\varepsilon _{{\rm{dep}}}} = 1.1{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
,
${\varepsilon _{{\rm{stow}}}} = 1.05{\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
,
${t_{{\rm{delay}}}} = 0$
s,
${t_{{\rm{dep}}}} = {t_{{\rm{stow}}}} = 2{T_c}$
and
${\rm{\Delta }} = 15{{\rm{\;}}^ \circ }$
, where
${T_c} = 17.3$
ms at this flight point. These spoiler parameters are the same as those depicted in Fig. 10, except
${\varepsilon _{{\rm{dep}}}}$
and
${\varepsilon _{{\rm{stow}}}}$
have been reduced. The very low
${\varepsilon _{{\rm{dep}}}}$
means that
${\varepsilon _{{\rm{spoiler}}}}$
exceeds
${\varepsilon _{{\rm{dep}}}}$
for a second time just before
$t = 3$
s. This initiates a rapid deployment and stowage, which in turn excites a high frequency (
$8.02$
Hz) bending mode of the wing. This causes further cycles of deployment and stowage, which quickly ‘lock in’ to the bending oscillations of the wing and the process becomes self-sustaining, reaching a steady state frequency of
$8.30$
Hz.
(a)
${{\rm{\varepsilon }}_{{\rm{spoiler}}}}$
and
${\rm{\delta }}$
vs time at
${{Z}} = 13,100$
m,
${\rm{M}} = 0.89$
and
${{H}} = 76.2$
m with
${{\rm{\varepsilon }}_{{\rm{dep}}}}$
reduced to
$1.1{{\rm{\varepsilon }}_{{\rm{spoiler}},1{\rm{g}}}}$
. The reduced
${{\rm{\varepsilon }}_{{\rm{dep}}}}$
causes the spoiler to initiate a limit cycle oscillation in the wing. (b) A gust encounter under the same conditions but with
${\rm{\Delta }}$
reduced to
${8^ \circ }$
. This reduces the influence
${{\bf{f}}_{{\rm{spoiler}}}}$
has over
${{\rm{\varepsilon }}_{{\rm{spoiler}}}}$
, which prevents oscillations from becoming self-sustaining. (c) Increasing
${{t}_{{\rm{stow}}}}$
to
$5{{{T}}_{\rm{c}}}$
also prevents oscillation, but without adversely affecting load alleviation performance.
Mitigations
The oscillations shown in Fig. 15(a) would clearly be undesirable in any practical implementation of the spoiler. As such, a number of strategies for mitigating these oscillations were investigated.
The root cause of the oscillation is that the application of
${{\bf{f}}_{{\rm{spoiler}}}}$
causes excessive changes in
${\varepsilon _{{\rm{spoiler}}}}$
, allowing oscillation to become self sustaining. One way to reduce this effect is to simply reduce
${\rm{\Delta }}$
, as this in turn reduces the maximum magnitude of
${{\bf{f}}_{{\rm{spoiler}}}}$
and thus the change in
${\varepsilon _{{\rm{spoiler}}}}$
caused by spoiler deployment. This is demonstrated by Fig. 15(b), which shows the same gust encounter as Fig. 15(a), only with
${\rm{\Delta }}$
reduced to
${8^ \circ }$
. Although some unwanted deployment of the spoiler remains around
$t = 3$
s, there is no longer sufficient coupling between
${{\bf{f}}_{{\rm{spoiler}}}}$
and
${\varepsilon _{{\rm{spoiler}}}}$
for the oscillation to become self-sustaining.
Naturally, reducing
${\rm{\Delta }}$
reduces the load alleviation performance of the spoiler (c.f. Fig. 13). However, oscillation can also be eliminated with only minimal impact on load alleviation performance by increasing
${t_{{\rm{stow}}}}$
whilst leaving
${t_{{\rm{dep}}}}$
unchanged. This scenario is shown in Fig. 15(c) where
${t_{{\rm{stow}}}}$
is increased from
$2{T_c} = 34.6$
ms to
$5{T_c} = 86.5$
ms, and
${\rm{\Delta }}$
returned to
${15^ \circ }$
. Extending
${t_{{\rm{stow}}}}$
relative to
${t_{{\rm{dep}}}}$
leads to an inherent asymmetry in the spoiler’s deployment and stowage profile. This asymmetry alters the phase of the deployment and stowage cycle relative to the wing’s vibration, ultimately reducing the amplitude of
${\varepsilon _{{\rm{spoiler}}}}$
oscillations such that they decay naturally. The stowage time of the spoiler concept investigated in our earlier work [Reference Wheatcroft, Mahadik, Groh, Pirrera and Schenk27] could readily be increased independently of the deployment time by introducing one-way damping to the mechanism.
Figure 16 shows
${\hat M_{3,{\rm{root}}}}$
for the spoiler with
${t_{{\rm{stow}}}} = 5{T_c}$
at the sizing flight point. Note that the sizing flight point is not the same as the one shown in Fig. 15. Comparison of Fig. 16 and Fig. 10(b) shows that extending
${t_{{\rm{stow}}}}$
independently of
${t_{{\rm{dep}}}}$
has no impact on the up-bend performance of the spoiler. As expected from the results in Fig. 14(b), extending
${t_{{\rm{stow}}}}$
has caused down-bend moments to decrease, though they are only marginally less than the baseline.
Other design considerations may also mean that spoiler oscillations are never encountered in practice. Oscillations were only observed when
${\varepsilon _{{\rm{dep}}}}$
was set very close to
${\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
, but some separation between
${\varepsilon _{{\rm{dep}}}}$
and
${\varepsilon _{{\rm{spoiler}},1{\rm{g}}}}$
is likely to be desirable to prevent the spoiler from deploying during regular aircraft operation as discussed above. Also, the slow increase in
${\varepsilon _{{\rm{spoiler}}}}$
after the initial peak is driven by a corresponding oscillation in
${\alpha _{{\rm{tot}}}}$
(c.f. Fig. 7), which may be damped out by the flight control system using the elevator.
${{\hat{M}}_{3,{\rm{root}}}}$
and
${\rm{\delta }}$
at the sizing flight point (
${{Z}} = 7,500$
m,
${\rm{M}} = 0.89$
and
${{H}} = 76.2$
m) for the spoiler with
${{t}_{{\rm{stow}}}}$
extended to
$5{{T}_{\rm{c}}}$
as per Fig. 15(c). Extending
${{t}_{{\rm{stow}}}}$
has not impacted the load alleviation performance of the spoiler, aside from a small decrease in down-bend moment.
It should also be noted that the low fidelity model used here may not be capable of accurately capturing the oscillation effect. The oscillations are heavily dependent on a high degree of coupling between spoiler aerodynamic loading
${{\bf{f}}_{{\rm{spoiler}}}}$
and spoiler input strain
${\varepsilon _{{\rm{spoiler}}}}$
, as well as the resonant frequencies of the wing. As discussed in Section 2.3, a stiffness-based beam model such as the one used here is unable to accurately represent the coupling between
${{\bf{f}}_{{\rm{spoiler}}}}$
and
${\varepsilon _{{\rm{spoiler}}}}$
, and indeed tends to magnify the effect. Consequently, higher fidelity modelling would be required in order to definitively establish whether or not spoiler oscillations are likely to be problematic in practice, and to further verify the mitigation strategies demonstrated here.
4.0 Conclusions
This work demonstrates that effective gust load alleviation can be achieved by a spoiler control surface that is actuated purely in response to wingbox strain [Reference Wheatcroft, Shen, Groh, Pirrera and Schenk26, Reference Wheatcroft, Mahadik, Groh, Pirrera and Schenk27]. The device investigated in this work is shown to be capable of reducing the peak wing-root bending moment experienced during a gust by up to
$17$
%, for the specific airframe considered herein. The performance of the spoiler is affected by its deployment and stowage threshold strains, maximum deployment angle and the overall time taken to deploy. Deploying the spoiler as early as possible during a gust is beneficial in alleviating peak up-bend moments. This can be achieved either by reducing deployment strain, or by shortening deployment time. Naturally, increasing the maximum deployment angle of the spoiler also reduces peak up-bend moments. Down-bend moments are often worsened by the spoiler, but they remain small compared to up-bend moments in all but a few permutations of flight point and spoiler design parameters. Spoiler configurations with a high stowage strain and fast stowage time have the least impact on down-bend performance, even improving performance in some cases. Repeated deployment and stowage can induce a limit cycle oscillation in the wing for a few particular flight points and spoiler configurations; however, it is not clear if this is a genuine physical phenomenon, or merely an artefact of the particular model used. In any case, these oscillations can be eliminated by reducing the influence of spoiler deployment on wing strain at the spoiler’s span station, or by slowing down spoiler stowage.
Future numerical work should focus on using higher fidelity models to more accurately capture variations in strain at the spoiler location due to spoiler deployment. Furthermore, improved aerodynamic modelling could be used to compute a more accurate relationship between spoiler deployment and spoiler loads. Ideally, this modelling would fully couple the mechanics of the nonlinear spoiler to that of the fluid. In addition, the effect of the spoiler on different airframes should be investigated, in particular those with more flexible, higher aspect ratio wings. Future experimental work will include validation of the strain-actuated spoiler concept on a flexible wind tunnel model with passively deploying spoilers.
Acknowledgements
E.D.W. is funded by an EPSRC Doctoral Training Partnership (DTP) studentship [Grant no. EP/T517872/1]. Parts of this work were carried out using the computational facilities of the Advanced Computing Research Centre, University of Bristol - http://www.bristol.ac.uk/acrc/http://www.bristol.ac.uk/acrc/.
Data availability statement
Data are available at the University of Bristol data repository, data.bris, at https://doi.org/10.5523/bris.1638ijt7qp36i2047npa80365h.
Competing interests
The authors declare none.
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