Effect of noise reducing components on nose landing gear stability for a mid-size aircraft coupled with vortex shedding and freeplay

Journal of Sound and Vibration 354 (2015) 91–103

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Effect of noise reducing components on nose landing gear stability for a mid-size aircraft coupled with vortex shedding and freeplay Petr Eret n, John Kennedy, Gareth J. Bennett Department of Mechanical and Manufacturing Engineering, Parsons Building, School of Engineering, Trinity College Dublin, Dublin 2, Ireland

a r t i c l e i n f o

abstract

Article history: Received 29 October 2014 Received in revised form 20 April 2015 Accepted 12 June 2015 Handling Editor: I. Lopez Arteaga Available online 2 July 2015

In the pursuit of quieter aircraft, significant effort has been dedicated to airframe noise identification and reduction. The landing gear is one of the main sources of airframe noise on approach. The addition of noise abatement technologies such as fairings or wheel hub caps is usually considered to be the simplest solution to reduce this noise. After touchdown, noise abatement components can potentially affect the inherently nonlinear and dynamically complex behaviour (shimmy) of landing gear. Moreover, fairings can influence the aerodynamic load on the system and interact with the mechanical freeplay in the torque link. This paper presents a numerical study of nose landing gear stability for a mid-size aircraft with low noise solutions, which are modelled by an increase of the relevant model structural parameters to address a hypothetical effect of additional fairings and wheel hub caps. The study shows that the wheel hub caps are not a threat to stability. A fairing has a destabilising effect due to the increased moment of inertia of the strut and a stabilising effect due to the increased torsional stiffness of the strut. As the torsional stiffness is dependent on the method of attachment, in situations where the fairing increases the torsional inertia with little increase to the torsional stiffness, a net destabilising effect can result. Alternatively, it is possible that for the case that if the fairing were to increase equally both the torsional stiffness and the moment of inertia of the strut, then their effects could be mutually negated. However, it has been found here that for small and simple fairings, typical of current landing gear noise abatement design, their implementation will not affect the dynamics and stability of the system in an operational range (F z r 50 000 N, V r 100 m/s). This generalisation is strictly dependent on size and installation methods. The aerodynamic load, which would be influenced by the presence of fairings, was modelled using a simple vortex shedding oscillator acting on the strut. The stability boundary was found to remain unaltered by vortex shedding. Significantly however, the addition of freeplay in the torque link was found to cause shimmy over the more typical operating conditions studied here. Unlike the no-freeplay case, there was a suppressed stabilising effect of increased torsional stiffness of the strut caused by the presence of fairing. No interaction between the vortex shedding and the freeplay on the stability threshold was observed. & 2015 Elsevier Ltd. All rights reserved.

n

Corresponding author. E-mail addresses: [email protected] (P. Eret), [email protected] (J. Kennedy), [email protected] (G.J. Bennett).

http://dx.doi.org/10.1016/j.jsv.2015.06.022 0022-460X/& 2015 Elsevier Ltd. All rights reserved.

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1. Introduction Airframe noise reduction is a significant task for the aircraft community in order to meet the strict objectives regarding noise reduction set for Europe. The relevant research effort and its directions can be found, for instance, in Dobrzynski et al. [6], Dobrzynski [7], and Li et al. [15]. During approach, when engines are operating at low thrust, the noise of the airframe contributes significantly to the overall noise signature of modern aircraft and may be a problem for people living around airports. One of the main sources of airframe noise is the landing gear [4]. Independent of aircraft type, the nose landing gear noise is generally measured to be a significant contributor to airframe noise [18]. Table 1 taken from Sijtsma [19] provides an example of ranking of the noise sources of A340 during approach (Fig. 1) with the nose landing gear as the loudest airframe noise source (engine noise excluded). Fairings or wheel hub caps can be installed to reduce some of this noise, see example in Fig. 2. A fairing is a noise reducing element whose function is mainly to deflect flow and produce a more aerodynamic and hence a quieter outline. Both inner and outer wheel hub caps cover wheel hub voids and hence the noise caused by the interaction of the air and the wheel may be reduced. Although these arrangements are relatively successful in wind tunnel measurements on model-scale [5,14], or even full-scale [4,8], they have performed poorly on real aircraft [9]. A landing gear is an inherently nonlinear and dynamically complex system. Any small design modification such as an application of a low noise solution can affect its behaviour. Shimmy is a well known low frequency vibration (10–30 Hz) of landing gear during ground manoeuvres. It can in principle be found on both nose and main landing gear, but occurrence of the latter is rare. Shimmy is usually not catastrophic, however it can contribute to pilot and passenger discomfort or eventually lead to accidents such as shown in Fig. 3. In this accident, incorrectly installed washers on the nose landing gear torque link component caused unscrewing and detachment of the hinge shaft nut. Free on its axle, the nose landing gear, shortly after touchdown, began to shimmy, which made the aircraft impossible to control, resulting in the collision of the nose landing gear with a concrete inspection pit, see report Bureau d'Enquêtes et d'Analyses (BEA) [2]. The causes of shimmy are numerous and Table 2 summarises some of them. A typical cause is mechanical freeplay in the torque link as studied by Sateesh and Maiti [17]. Strikingly, most of the causes can be attributed to a negligence in maintenance or to an inadequate design. The nose landing gear of a typical mid-size commercial passenger aircraft is of a dual wheel configuration and has four main vibrational modes: a torsional mode corresponding to the rotation about the strut axis, a lateral mode that is representative of vibrations of the gear about an axis passing through the fuselage centreline, a vertical mode associated with the shock dampers of the gear (which are generally called oleos in the context of aircraft landing gears) and a longitudinal mode, in which the gear may bend forwards and backwards in a straight-line travel of the direction of the aircraft. Thota et al. [25] have found that the longitudinal mode does not actively participate in the nose landing gear dynamics over the entire range of forward velocity and vertical force (both crucial parameters are introduced later). Similarly, the vertical mode of a commercial aircraft is generally sufficiently damped so that it is not excited on today's smooth runways or taxiways. The remaining two modes of vibration are coupled via the tyre-ground interaction and play an important role in the occurrence of shimmy in aircraft. An aircraft nose landing gear generally features a nonzero rake angle and its torsional mode is very strongly damped. Nevertheless, shimmy oscillations still occur in aircraft landing gears, and this has been studied mostly numerically, see Somieski [21], Terkovics et al. [24], Thota et al. [26,22], for example.

Table 1 Noise sources of A340 during approach; the A-weighted sound levels were summed across all frequency bands and averaged over all emissions angles [19]. Component

Average power (dBA)

Engine 3 exhaust Engine 2 exhaust Engine 1 exhaust Engine 4 exhaust Nose landing gear Left main gear Right main gear Tail 2 Engine 2 inlet Engine 3 inlet Flap edge 4 Middle main gear Engine 1 inlet Engine 4 inlet Right slat horn Left slat horn Flap edge 1 Flap edge 3 Flap edge 2 Engine 4 vane

131.12 130.49 130.40 129.91 128.51 127.87 127.86 126.03 125.73 125.55 125.34 125.14 125.10 124.98 124.74 124.71 124.22 124.02 123.91 123.71

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Fig. 1. Acoustic image of A340 at 2000 Hz (1/3 octave band) with source power integration areas [19].

Fig. 2. Example nose landing gear concept with (a) installed fairings on lower arm joint, steering system and tow fitting and (b) wheel hub caps.

Fig. 3. Airbus A300-B4 Accident due to shimmy, Bratislava Airport, November 2012, from Bureau d'Enquêtes et d'Analyses (BEA) [2].

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Table 2 Selected causes of shimmy. Design

Maintenance

Other

Insufficient overall torsional stiffness of the gear Inadequate trail (positive trail reduces shimmy) Flexible suspension Low torsional stiffness of the strut

Excessive mechanical freeplay (torque link, wheel axle, Rough surface irregularities etc.) Incorrectly re-installed/assembled components Interaction with fuselage (strong side wind, etc.) (Unevenly) worn parts (bearings, shimmy damper, etc.) Uneven tyre pressure Unbalanced wheels

Fig. 4. Nose landing gear schematic adapted from Thota et al. [24,26].

In this paper, the nonlinear multi degree of freedom model of dual wheel nose landing gear developed by Thota et al. [26] is applied to estimate the influence of noise reducing components on nose landing gear stability. This is achieved through an increase of relevant model parameters to address a hypothetical effect of additional fairings and wheel hub caps. The presence of fairings also requires an investigation into their influence on the aerodynamic load of the system. As nose landing gear assembly has reasonably high values of mechanical parameters (stiffness, damping), motion dependent fluid forces, which have usually crucial effect on the stability of weakly damped, light systems such as tube array in heat exchanger (see example in [10]), are not considered. An effect of motion independent fluid forces in a form of vortex shedding oscillator acting on the strut in the lateral mode is modelled. In addition, a backlash is implemented into the model to study a scenario with mechanical freeplay in the torque link and to investigate its potential interaction with the vortex shedding especially.

2. Mathematical model Fig. 4 provides a schematic of a nose landing gear, where the positive X-axis points towards the rear direction of the aircraft, the Z-axis is the upward normal to the (flat) ground, and the Y-axis completes the right-handed coordinate system. The nose landing gear of an aircraft consists of a strut that is attached to the aircraft fuselage and coupled to the ground via one or more wheels with flexible tyres. The strut has a torque link providing a firm connection between an upper part of the strut and a lower cylinder. The strut is able to rotate about its axis S, which gives rise to a steering angle ψ. The wheel axle, offset from the strut axis by a mechanical trail (caster) of length e, supports the wheels with a tyre of radius r. Importantly, the strut axis is inclined to the vertical at a rake angle ϕ. The presence of a nonzero rake angle ϕ is incorporated into the model and it has three geometrical effects in an aircraft nose landing gear. They are briefly introduced below and more details can be found in Thota et al. [23]: 1. It induces an effective caster length eeff defined by eeff ¼ e cos ϕ þ ðr þ e sin ϕÞ tan ϕ

(1)

2. For a nonzero rake angle the swivel angle θ of the wheel with the ground is different from the steering angle ψ; namely, it is given by θ ¼ ψ cos ϕ. 3. There is a tilt γ ¼ ψ sin ϕ of the wheel when the steering angle ψ is nonzero.

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Table 3 System parameters used in model. Symbol

Parameter

Value

e lg kψ

Structure Caster length Gear height Torsional stiffness of strut

0.12 m 2.5 m

cψ Iψ kδ

Torsional damping of strut Moment of inertia of strut Lateral bending stiffness of strut

cδ Iδ ϕ D d

Lateral damping of strut Moment of inertia of strut Rake angle Distance between the wheel centres Diameter of the strut Tyre Radius of nose wheel Contact patch length Damping coefficient of elastic tyre Self-aligning coefficient of elastic tyre Restoring coefficient of elastic tyre Vertical stiffness of elastic tyre

r h cλ kα kλ kv σ αm I Fz V ρ

Relaxation length Self-aligning moment limit Wheel moment of inertia Operational Vertical force on the gear Forward velocity Density of air

3:8n105 N m rad  1 300 N m s rad  1 100 kg m2 6:1n106 N m rad  1 300 N m s rad  1 600 kg m2 0.1571 rad (91) 0.3 m 0.15 m 0.362 m 0.1 m 270 N m2 rad  1 1.0 m rad  1 0.002 rad  1 4n106 N m  1 0.3 m 0.1745 rad (101) 1.1 kg m2 up 50 n 103 N up 100 m s  1 1.2 kg m  3

Both wheels, which are symmetrically mounted on the axle with a separation distance D between their centres, are assumed to move independent of each other (unlike in the case of co-rotating gears where the wheels are fixed to the axle). There is now a tyre equation for each wheel, where we assume that the tyre characteristics, including the inflation pressure, are identical for both tyres. Finally, each wheel has the moment of inertia I about its spinning axis. The overall model for the torsional and lateral vibrational modes and their coupling through the elastic tyres of both wheels then takes the form of a six-dimensional system of ordinary differential equations of the first order for the two modes and the kinematic equations of the nonlinear tyres; all variables are defined in Table 3 and following sections. The torsional mode ψ (Eq. (2)) and the lateral mode δ (Eq. (3)) are strongly coupled via the lateral deformations λL;R of the left and right tyres taken at the front of the tyre contact patches (Eqs. (4)–(5)). These equations are obtained using adaptation of the model developed by von Schlippe and Dietrich [28], wherein the deflected shape of each tyre is represented by a stretched string. The region where the string is in a contact with road has a length of h (i.e. contact patch length) as depicted in Fig. 4 and can be approximated by a constant [26]. In general, accurate measurement of tyre characteristics is rather difficult, however, some experimental data and contact patch length prediction can be already found in Smiley and Horne [20]. The aircraft body is modelled as a block of mass exerting a vertical force Fz on the gear, which is moving at a fixed horizontal velocity V in the negative X-direction as shown in Fig. 4. The individual, unmodified terms have already been described in Thota et al. [24,26], however, for the sake of clarity they are detailed in the following sections prior to the presentation of the additional and modified terms. The simulation parameters for the nose landing gear geometry and the tyre characteristics used in this study are provided in Table 3. Some of these are taken from Thota et al. [26] and others are changed so that those tabulated here are realistic values of the same order of magnitude as the generic values for a mid-size passenger aircraft: I ψ ψ€ þM K ψ þ MDψ þ M K αL þM K αR þM DλL V þ M DλR þ2I δ_ þðF K λL þF K λR Þeeff r
 D  F zL sin ϕ eeff sin θ þ cos θ þlg sin δ 2
 D  F zR sin ϕ eeff sin θ  cos θ þlg sin δ ¼ 0 2 V I δ δ€ þM K δ þM Dδ þM λδL þM λδR  2I ψ_ r
D  F zL eeff sin θ þ cos θ þ lg sin δ 2
 D  F zR eeff sin θ  cos θ þlg sin δ ¼ 0 2

(2)

(3)

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V

λ_L þ λL  V sin θ  lg δ_ cos δ σ D  ðeeff hÞψ_ cos θ cos ϕ  ψ_ sin θ cos ϕ ¼ 0

(4)

2

V

λ_R þ λR  V sin θ  lg δ_ cos δ σ D  ðeeff hÞψ_ cos θ cos ϕ þ ψ_ sin θ cos ϕ ¼ 0

(5)

2

2.1. Torsional mode of the landing gear The moment M K ψ due to the torsional stiffness of the strut and moment MDψ due to the torsional damping of the strut are expressed, respectively, by M K ψ ¼ kψ ψ

(6)

M Dψ ¼ cψ ψ_

(7)

where kψ and cψ are the torsional stiffness and damping coefficients of the strut, respectively. The self-aligning torsional moment M K α about the strut axis is expressed by [21] 8 α   < kα m F zL=R sin αL=R αm ifðjαL=R j r αm Þ π π MK αL=R ¼ :0 ifðjαL=R j 4 αm Þ

(8)

where kα is the self-aligning moment coefficient and αm is the self-aligning moment limit. The overall vertical force Fz on the nose landing gear is asymmetrically divided into two forces F zL and F zR on the left and right wheels:
   Fz kv D 18 F zL=R ¼ sin γ þ δ (9) Fz 2 where kv is the vertical stiffness of the tyre for a given inflation pressure. The width of the tyres also produces a stabilising effect on the nose landing gear structure due to their lateral damping characteristics given by 1 M DλL ¼ MDλR ¼ cλ ψ_ cos ϕ V

(10)

where cλ is the lateral damping coefficient of the tyre. The gyroscopic effect due to the forward velocity V and rate of change of torsion of the nose landing gear is expressed by moment 2I δ_ V=r. And finally, the restoring (cornering) force of each tyre is defined by [24]   F K λ ¼ kλ tan  1 7 tan αL=R L=R     cos 0:95 tan  1 7 tan αL=R F zL=R (11) where kλ is the cornering force coefficient and the slip angles αL=R of each tyre are related to the corresponding tyre deformations λL=R by Eq. (12) with σ as the relaxation length of the tyre:
 λ αL=R ¼ tan  1 L=R (12)

σ

2.2. Lateral bending mode of the landing gear The moment M K δ due to the lateral bending stiffness of the strut and moment M Dδ due to the lateral damping of the strut are expressed, respectively, by M K δ ¼ kδ δ

(13)

MDδ ¼ cδ δ_

(14)

where kδ and cδ are the lateral bending stiffness and damping coefficients of the strut, respectively. The moments MλδL and M λδR describe the coupling between the vibrational modes and they are given by M λδL=R ¼ lg F K λL=R cos θ cos ϕ

(15)

where lg is the distance between the ground and the nose landing gear attachment point in the fuselage along axis S.

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The gyroscopic effect due to the forward velocity V and rate of change of lateral bending of the nose landing gear is expressed by moment 2I ψ_ V=r. 2.3. Tyre kinematics The equations of motion of the tyres of the left (Eq. (4)) and right wheel (Eq. (5)) are based on the well established stretched string model. They are identical except for the second-order terms 7ðD=2Þψ_ sin θ cos ϕ. This higher-order difference between the tyre equations, which is usually neglected by some researchers Pacejka [16] and Van Der Valk and Pacejka [27] in their simplified models, enables to deal with motion over any slip angle. 2.4. Vortex induced vibration Vortex shedding is a known mechanism responsible for flow induced vibration of a body subject to an oncoming flow. A simple wake oscillator model can be applied to the strut and the lift force Fs acting in the lateral direction is defined by   F s ¼ 12 ρU 2 LdC L sin 2π f VS t (16) where ρ ¼ 1:2 kg=m3 is the air density, U (¼V) is the freestream velocity, L (  lg ) is the length of strut, d¼0.15 m is the diameter of strut, CL is the amplitude of lift coefficient. For a complex system such as landing gear, the lift coefficient can be varied on the basis of geometry or element location, but in this study a typical value for a cylinder CL ¼0.3 [1], is considered even though this might be large as the strut lift force consists solely of shear forces, which are relatively small. The frequency of vortex shedding is f VS ¼ ðU=dÞSt cos ϕ, where St¼0.2 is the Strouhal number [1]. If we assume that the lift force Fs acts approximately at the distance lg =2 then it generates a moment M s ¼ F s lg =2, which is included into a modified Eq. (3) in this work. 2.5. Torsional freeplay The moment M K ψ due to the torsional stiffness of the strut with a freeplay has the form of [11] 8
   > < kψ ψ 1  Δ sgn ψ  Δ if ðjψ j4 ΔÞ ψ MK ψ ¼ > : 0 if ðjψ jr ΔÞ

(17)

where Δ is the freeplay angle. This definition means that a non-smooth transition to linear stiffness is considered at the boundaries of the freeplay region. Fig. 5 shows the trend of moment due to the torsional stiffness with freeplay Δ ¼ 0:11 and kψ ¼ 3:8n105 N m=rad. 3. Results 3.1. Bifurcation analysis [26] Fig. 6 depicts a two-parameter bifurcation diagram in the (V; F z ) plane obtained by Thota et al. [26] for D¼0.1 m and I ¼0.1 kg m2. The vertical force Fz and forward velocity V are two of the most important parameters defining conditions of aircraft taxi, take-off and landing. The (V; F z ) plane is output from bifurcation analysis to characterise the observable behaviour of the system. The starting point of this systematic investigation is the transition to the shimmy oscillation. The bifurcation curves are used to bound regions of different behaviour. In order to show the full bifurcation curve loops, the diagram exceeds a typical operational range of a mid-size passenger aircraft, which is F z r50 000 N and V r 100 m=s. In this paper, the equations of motion are solved numerically in MATLAB (ode45 function) using Runge–Kutta formulas with a fixed time step of 1/1024 s, non-zero initial conditions in lateral mode (δ0 ¼ 0:001) and the total integration time of one minute. The obtained results are inferred from the features of time domain simulations of the full nonlinear model. Based on the parameters specified in Table 3, the performance of the nonlinear model has been verified on the stability thresholds for velocities V lower than 100 m/s as shown in Fig. 6. 3.2. Limit structure parameters A simple sensitivity analysis of structural parameters has been performed for situations when system torsional instability reaches the boundary of typical operating conditions (i.e. the upper limit of Fz ¼50 000 N is a solid green line in Fig. 7). Fig. 7 shows several examples of the limit situation compared to the baseline design parameters of I¼1.1 kg m2 and D¼0.3 m, which are more realistic than in the previous example in Fig. 6 (the area under each individual threshold curve represents no shimmy solutions). The overall comparison is provided in Table 4. The parameter with the highest sensitivity is torsional damping cψ . Only a small decrease of 11 percent in torsional damping reduces the system stability boundary to the limit of F z ¼ 50 000 N. In fact, all parameters associated with the torsional mode, when modified within 7 30 percent range, can cause shimmy, which may be observed over some typical operating conditions. Should a fairing significantly affect these structural

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3000

2000

[Nm]

1000

M

0

−1000

−2000

−3000

−0.5

0

0.5

Ψ [°] Fig. 5. Moment due to the torsional stiffness with freeplay, Δ ¼ 0:11 and kψ ¼ 3:8n105 N m/rad. 5

x 10 6

5

F [N]

4

3

2

1 current study 0

20

40

60

80

100

120

140

160

180

V [m/s] Fig. 6. Bifurcation diagram adapted from Thota et al. [26]. Figure features an unshaded region of no shimmy, right-slanted shading for stable torsional shimmy oscillations, left-slanted shading for stable lateral shimmy oscillations and chequered region for bi-stability. The identified bifurcation curves are Hopf bifurcations of the torsional mode Ht and the lateral mode Hl together with torus bifurcation curves T t ; T l and a saddle node of limit cycle bifurcation curve SL. Different bifurcation curves meet at double-Hopf bifurcation points HH and degenerate Hopf bifurcation points DH.

parameters, then installation of a low noise solution would be unfavourable. On the other hand, the lateral bending stiffness of strut kδ and the moment of inertia of the strut I δ are less influential, moreover the lateral damping of the strut cδ is found to be completely insensitive for merging the Hopf bifurcation curve Ht with the upper limit of typical operating conditions.

3.3. Low noise solutions In general, the overall landing gear weight is about 3–5 percent of the maximum take-off weight (MTOW). A generic landing gear assembly weight breakdown can be found in Currey [3], and for example, a nose landing gear rolling stock (wheels, tyres) is 2 percent of total landing gear weight. Although the landing gear weight is relatively small, it is assumed that aircraft manufacturers would be unwilling to allow the installation of unnecessarily heavy and complicated additional structures with many attachment points to the existing landing gear. This anticipates that the addition of low noise solutions would not dramatically affect design parameters. For example, installation of sheet aluminium inner and outer wheel hub caps (radius of 0.18 m, thickness of 8 mm) on one wheel (r ¼0.362 m) results in an increase of the wheel moment of inertia I by nearly 7 percent with respect to the typical value of 1.1 kg m2 (Table 3). The impact of a fairing on system parameter values cannot be exactly estimated unless a specific design solution is discussed. In order to investigate the quantitative

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4

10

x 10

9

z

F [N]

8

7

6

5

4

0

20

40

60

80

100

V [m/s] Fig. 7. Stability diagram with limit situations. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.) Table 4 Limit structure parameters. Parameter

Design value

Limit value

Difference

D cψ kψ

0.3 300

0.475 268

3:8n105 100 0.12 2.5 91 300

2:93n105 127 0.096 8.375 4:31 na

þ58%  11%  23%

6:1n106 600

2:93n105 1284

Iψ e lg ϕ cδ kδ Iδ

þ27%  20% þ235%  52% na  46% þ114%

effect of noise reducing components on the system behaviour, the relevant parameters, detailed below, are therefore deliberately exaggerated by 10 percent. It was revealed that the torsional damping cψ has a high impact on the system stability. If there is an increase of the torsional damping cψ by a fairing, then its installation has a stabilising effect. However, no significant influence of fairings on the torsional damping is considered in this study. This is explained by the fact that fairings are usually small and simple devices. In order to increase the torsional damping cψ a sophisticated device such as magneto-rheological fluid-based damper must be used as reported by Sateesh and Maiti [17]. The structural parameter, which can be primarily effected by fairings, is the moment of inertia of the strut I ψ . Results in the previous section suggest that the increase of this parameter by a fairing has a destabilising effect. Fairings can also influence the torsional stiffness of strut kψ . A change to the overall torsional stiffness of the strut will depend on the actual attachment method, but an increase of torsional stiffness kψ by 10 percent only is considered in the simulations. The increase of torsional stiffness kψ proved to be anti-shimmy. Interestingly, a mutual combination of I ψ and kψ , both increased by 10 percent, is negated. Hence if potentially a fairing is attached to the existing structure, influencing equally both torsional stiffness and moment of inertia, the stability threshold of the system can remain unchanged. The results are summarised in Fig. 8. The wheel hub caps principally change the wheel moment of inertia I. The increase of this parameter has a minimal impact on the system stability boundary as depicted in Fig. 9. The effect of wheel moment of inertia I has also been previously studied by Thota et al. [26] for D ¼0.1 m and I ranging from 1.1 to 1.3 kg m2. In principle, the increased values of moment of inertia I did not change the stability boundary for less than V¼100 m/s, which agrees with the current findings. The results presented in Fig. 9 also confirm the conclusion made by Thota et al. [26] that the torsional mode is less stable with increasing wheel separation distance as the stability threshold moves lower in the (V; F z ) plane. For the sake of study completeness, Fig. 9 shows that the increase of parameters I δ and kδ has the same effect as the increased value of moment of inertia I. Given that computational parameters have been deliberately exaggerated, in the absence of freeplay, the results generally demonstrate that additional fairings and wheel hub caps are unlikely to influence nose landing gear dynamics and

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4

10

x 10

9

F [N]

8

7

6

5

4

0

20

40

60

80

100

V [m/s] Fig. 8. Stability diagram with low noise solutions: effect of I ψ and kψ .

10

x 10

9

z

F [N]

8

7

6

5

4

0

20

40

60

80

100

V [m/s] Fig. 9. Stability diagram with low noise solutions: effect of I; I δ and kδ .

stability as seen from the point of view of the steady-state analysis. This conclusion is made cognisant of the aforementioned caveats. It should be also mentioned that the problem of NLG behaviour is simplified into the model developed by Thota et al. [26] and the main relevant physical effects of the low noise solution might not be truly addressed. However, it is believed that these solutions can be sufficiently approximated by the increase of the system parameters of the model, especially the additional moment of inertia of the strut.

3.4. Vortex shedding No impact of vortex shedding on the stability boundaries was found across a typical operational range as shown in Fig. 10. The vortex shedding as modelled here represents an autonomous excitation uncoupled to the system response. For a linearised system the stability is governed by the system parameters only. As the response of complete nonlinear system is investigated at the stability thresholds, the study shows that the system response is not affected by the vortex shedding especially when vortex shedding frequencies coincide with modal frequencies of the nose landing gear assembly during aircraft slow-down. The presence of fairings would potentially modify the lift coefficient of nose landing gear, but again the simulations proved the insensitivity to variations of the lift coefficient (Fig. 10). Also a strong side wind should pose no problem for the model as it stands, as the associated vortex shedding forces which would result act in the longitudinal direction the mode of

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4

10

x 10

9

z

F [N]

8 I = 1.1 kg m2, D = 0.3 m cL = 0.3

7

cL = 1 6

5

4

0

20

40

60

80

100

V [m/s] Fig. 10. Stability diagram with vortex shedding. 4

10

x 10

9 8 7

z

F [N]

6 5 4 3

freeplay Δ = 0

2

freeplay Δ = 0.1

1

freeplay Δ = 0.1°, c L = 1

0

° °

freeplay Δ = 0.1°, c L = 0.3

0

20

40

60

80

100

V [m/s] Fig. 11. Stability diagram with freeplay. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

which is not actively participating in dynamics. A future work could investigate a more interesting scenario of interaction between the fuselage and the nose landing gear in the presence of wind during aircraft ground manoeuvres.

3.5. Freeplay The work by Howcroft et al. [12] indicates that the translational freeplay of the order of 100 mm introduced at the torque link apex produces an angular freeplay of the order of 10  1 degrees. Fig. 11 shows the effect of freeplay Δ ¼ 0:11 for the baseline design parameters (I ¼1.1 kgm2, D¼ 0.3 m). The system is found to be unstable over more typical operating conditions (below Fz ¼50 000 N – solid green line). From all model parameters the strong nonlinearity of freeplay has the most crucial effect on system dynamics and appropriate attention should be paid to the freeplay tolerances during maintenance of landing gear. No interaction between the torsional freeplay and the vortex shedding has been observed and the stability boundary for freeplay remains unchanged as depicted in Fig. 11. Moreover, when fairings are taken into account, the increased moment of inertia of the strut I ψ , together with freeplay, tends to destabilise the system and the recovery effect of increased torsional stiffness of the strut, which was obvious in the no-freeplay case, is suppressed as shown in Fig. 12, indicating that the addition of fairings tend to exacerbate the problem.

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0

5

10

15

20

25

ψ

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V [m/s] Fig. 12. Stability diagram with freeplay and fairing.

Fig. 13. A time record of typical impact load on landing gear [13].

4. Discussion The technique exploited in the previous sections allows a detailed systematic investigation, however, it requires the transients to settle down. During typical landing and take-off scenarios there is not sufficient time for these transients to settle down to the limiting behaviour due to the variations of velocity V and loading force Fz. In order to capture a true effect of transients it would be important to know how the landing gear system behaves in the (V; F z ) plane. There might be a scenario when considerable amplitude and the time duration of these transients are achieved leading eventually to some other complications. Fig. 13 shows a typical impact load on both main and nose landing gear taken from Ladda and Struck [13] (the authors did not provide axes scales). It is noteworthy that approximately 40 percent of the static load (Fz – bottom of Fig. 13) is firstly applied on the main landing gear. This is because the wings still provide a large lift force. As the rolling aircraft reduces velocity, the lift is decreased gradually until total aircraft weight presses on the landing gear. This is obvious for both main and nose landing gear. The nose landing gear load peaks initially at the same level as the load during rollout. Unfortunately the information about system velocity is not available to perform a further analysis of aircraft landing gear behaviour, but several scenarios of landing could be modelled in a future parametric study. The work of Thota et al. [26] and the present extension showed some interesting results at the theoretical level. An experimental verification of the essential findings is a following logical step, however associated test campaign costs would not be low. This particularly explains why there is no available experimental database up to date. Moreover, aircraft and landing gear manufacturers wish to identify completely the key system elements that may cause shimmy before any experimental campaign on model or full scale model is considered for a particular configuration of aircraft. Therefore, numerical simulations of more complicated models (including main landing gear, fuselage and wind directions) can be expected to be performed. 5. Conclusion The paper investigates the nose landing gear ground dynamics of a mid-size aircraft affected by low noise solutions such as fairings and wheel hub caps in a typical operational range (F z r 50 000 N, V r 100 m/s). A previously published nonlinear multi degree of freedom system is used as a starting point to describe the dynamics of the torsional mode, the lateral mode and their tyre-ground coupling. The noise reducing components are modelled by an increase of the relevant model structural parameters to address a hypothetical effect of additional fairings and wheel hub caps. In addition, aerodynamic load, which

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would be influenced by the presence of fairings, is modelled using a simple vortex shedding oscillator acting on the strut in the lateral mode as flow forces in the longitudinal mode do not contribute to the dynamics of system. A mechanical freeplay in the torque link and its possible interaction with vortex shedding and fairings is also considered. Results showed that the wheel heel hub caps are not a threat to stability. The installation of a fairing can have a destabilising effect due to the increased moment of inertia of the strut and stabilising effect if its installation leads to an increase in the torsional stiffness of the strut. If the fairing increases equally both the torsional stiffness and the moment of inertia of the strut, then their effects can be mutually negated. in situations where the fairing increases the torsional inertia with little increase in the torsional stiffness, a destabilising effect can result. The aerodynamic load on the nose landing gear studied here has no influence on the stability boundary. In contrast the addition of freeplay in the torque link has a crucial effect on the stability threshold and caused shimmy over the more typical operating conditions examined in this work. There was a suppressed stabilising effect of increased torsional stiffness of the strut caused by the presence of fairing and the freeplay. The stability boundary for the freeplay case was not modified by the vortex shedding. Acknowledgements This work was partly supported by the WENEMOR project, Grant agreement no: 278419, and the ALLEGRA project, Grant agreement no: 308225 funded by the European Union FP7 CleanSky Joint Technology Initiative. References [1] R.D. Blevins, Flow-Induced Vibrations, Van Nostrand Reinhold, New York, 1990. [2] Bureau d'Enquêtes et d'Analyses (BEA), Lateral Runway Excursion During Landing Roll, Nose Landing Gear Collapse, Technical report, ei-c121116.en, 2013. [3] N.S. Currey, Aircraft Landing Gear Design: Principles and Practices. AIAA Education Series, American Institute of Aeronautics and Astronautics, Washington, 1988. [4] W. Dobrzynski, H. Buchholz, Full-scale noise testing on Airbus landing gears in the German-Dutch wind tunnel, 3rd AIAA/CEAS Aeroacoustics Conference, AIAA paper 97-1597-CP, 1997. [5] W. Dobrzynski, L.C. Chow, M.G. Smith, A. Boillot, O. Dereure, N. Molin, Experimental assessment of low noise landing gear component design, International Journal of Aeroacoustics 9 (6) (2010) 763–786. [6] W. Dobrzynski, R. Ewert, M. Pott-Pollenske, M. Herr, J. Delfs, Research at DLR towards airframe noise prediction and reduction, Aerospace Science and Technology 12 (1) (2008) 80–90. [7] W.M. Dobrzynski, Almost 40 years of airframe noise research: what did we achieve? Journal of Aircraft 47 (2) (2010) 353–367. [8] W.M. Dobrzynski, B. Schöning, L.C. Chow, C. Wood, M. Smith, C. 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