Research ArticleExperimental Study of Aircraft Achieving Dutch Roll ModeStability without Weathercock Stability

Jingcheng Fu ,1 Jun Huang ,1 Lei Song ,1 and Daqing Yang2

1School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China2Centre of Aeronautics, AIRC, Cranfield University, Bedford MK430AL, UK

Correspondence should be addressed to Lei Song; songlei@buaa.edu.cn

Received 11 May 2019; Accepted 9 October 2019; Published 11 February 2020

Academic Editor: Paolo Castaldi

Copyright © 2020 Jingcheng Fu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Weathercock stability is usually considered essential to achieve normal flight, while the Dutch roll mode stability can still beachieved without weathercock stability which has been algebraically proved. This paper proposed a flight experiment toinvestigate the characteristics of an airplane with Dutch roll mode stability but no weathercock stability. Firstly, the algebraicanalysis based on a standard lateral-directional mode approximation was made to demonstrate the effect of yawing stabilityderivative Cnβ on the Dutch roll mode characteristics. The flight experiment was organized after that using a model glider whichwas modified to have zero Cnβ but with marginal change on Cyβ. The convergence of Dutch roll mode in flight meets thealgebraic and numerical analysis as expected. However, the difference of handling characteristics between the original andmodified configurations indicates some other roles the weathercock stability plays in flight as well as some limitations ofutilizing mode criterion in flight quality analysis.

1. Introduction

Weathercock stability, also known as directional static stabil-ity or yaw stiffness, is the ability of the aircraft to turn in towind in order to maintain directional equilibrium. The staticstability principle suggests that weathercock stability is essen-tial for a stable airplane [1]. The derivative Cnβ that corre-sponds to the yawing moment due to sideslip is designed tobe a positive value for most conventional airplanes as a resultof weathercock stability assumption. The main contributionto Cnβ comes from vertical stabilizer. The fuselage and wingare also major contributors to the value of this derivative [2].

From the perspective of lateral-directional stability studyin flight dynamics, the perturbed state problems were firstintroduced in 1911 to investigate whether the response tothe external disturbance was acceptable [3, 4]. After the air-plane stability quartic in terms of its polynomial coefficientswas rooted by Bairstow, the Routh stability criterion wasthe first used to study the dynamic stability behavior of anairplane; then, the term Dutch roll was used by Hunsakerto identify the oscillatory roots of the lateral-directional char-

acteristic equations [5–8]. For the lateral-directional dynamicstability, a hypothetic small airplane with various modifi-cations of the fin area and dihedral setting was used toinvestigate the motions caused by the lateral disturbancebefore 1938 [9]. As demonstrated in Ref. [9], with differentcombinations of Cnβ and Clβ, types of spiral and oscillatorydivergence limits on these two derivatives were found.

To make specifications of the flight quality for newmilitary airplanes, MIL-8785B (ASG), which includes spec-ifications of lateral-directional mode characteristics, waspublished in 1969 based on the data from typical existing air-crafts [10]. It was continuously validated and revised by theUS military, institutes, and manufacturers; then, the finalrevision of MIL-F-8785C was issued in 1980 [11]. Later in1995, this military specification was expanded and revisedto MIL-STD-1797A and designated as a handbook for guid-ance only, where the flight quality specifications for thelateral-directional mode were involved identically [12].

For the lateral-directional stability and control studyfrom the designer’s perspective, in the preliminary configura-tion design stage, the tail-volume method is often used in the

HindawiInternational Journal of Aerospace EngineeringVolume 2020, Article ID 8971275, 8 pageshttps://doi.org/10.1155/2020/8971275

first-cut estimation of vertical stabilizer sizing. According tothe statistics of the tail-volume range for specific categoriesof existing aircrafts, the appropriate tail volumes are deter-mined, and the area of the vertical stabilizer can be obtained[2, 13]. As suggested in Ref. [14], for most airplanes, thederivative Cnβ has a lower limit of 0.001-0.0025.

The vertical stabilizer appears indispensable for an air-craft. However, in aviation history before the stability aug-ment system was invented, several aircrafts with flying wingconfigurations such as Ho.229 and N-9M appeared whichhave no vertical stabilizer to provide enough directional staticstability. On the other hand, it has been proven that thelateral-directional stability of this configuration can beachieved by adjusting the spanwise dihedral layout withoutinstalling a stability augment system [15, 16]. Instead offocusing on the required lateral-directional stability deriva-tive Clβ or Cnβ, a criterion of dynamic stability was includedin the aerodynamic design for flying wing, which consideredthe Dutch roll mode characteristics as the critical element forflight qualities [17]. Moreover, a unified algebraic approachwas developed to approximate aircraft lateral-directionalmodes and predict departure susceptibility [18]. And inRef. [19], the relationship between vertical stabilizer andlateral-directional stability was studied and illustrated bysome representative experiments. Above all, the study on fly-ing wing aircrafts and the Dutch roll mode stability charac-teristics illustrates the possibility of normal flight with nosufficient weathercock stability. But the role that weathercockstability played in normal flight is still worth investigation.

This paper is aimed at demonstrating that the weathercockstability is not essential to achieve Dutch roll mode stabilityfor a conventional layout airplane but has its influence onflight quality. The demonstration is made from three aspects:algebraic analysis, numerical study, and flight experiments.The flight test result may indicate some other significanceof the derivative Cnβ in flight quality which are not demon-strated in Dutch roll mode flight quality specifications.

2. The Essential Factor for Dutch RollMode Stability

To demonstrate the weathercock stability unessential andfind the crucial factors to achieve the Dutch roll mode stabil-ity, the classic lateral-directional mode literal approximationwas considered for theoretical research. In fact, such literalexpression involves aerodynamic and inertial data and itprovides valuable insight into the connection between therelative values of certain key derivatives and behavior of air-craft motion [18].

Commonly, the derivation of the approximation con-tains omissions of relatively unimportant terms, to makethe relationship between the cause and effect apparent. Butfor the purpose of studying the crucial factor for Dutch rollstability, some certain omissions need examination again inthe developing of approximation. The simplified model forthe approximation is obtained by separating slow modefrom fast modes. That is, for lateral-directional motions,the fourth-order equation approximation is made by sepa-

rating the spiral mode (first-order system) from the rolland Dutch roll mode (third-order system).

To develop the Dutch roll mode natural frequencyapproximation, we designate the stability axis referenceframe and assume a condition of level flight; the dynamicmatrix from the lateral-directional linearization small distur-bance equations is given by [20]

Alat =

Yβ/V0 0 −1 g/V0

Lβ′ Lp′ Lr′ 0

Nβ′ Np′ Nr′ 0

0 1 0 0

2666664

3777775, ð1Þ

where Li′and Ni′are standard primed derivatives, V0 refers totrimmed level flight velocity, and g refers the gravitationalacceleration. Consider the standard factorization of the char-acteristic polynomial for lateral-directional modes,

Plat sð Þ = s + λSð Þ s + λRð Þ s2 + 2ζDRωDRs + ω2DR

� �: ð2Þ

The subscripts S,R,DR refer to the spiral, roll, and Dutchroll modes, respectively. Letting

Plat sð Þ = s4 + Bs3 + Cs2 +Ds + E, ð3Þ

one has

B = λR + λS + 2ζDRωDR,C = ω2

DR + 2ζDRωDR λR + λSð Þ + λRλS,D = λR + λSð Þω2

DR + 2ζDRωDRλRλS,E = λRλSω

2DR:

ð4Þ

The fourth-order polynomial coefficients can be deter-mined by derivatives in matrix Alat. Consider the fourcoefficients in the characteristic polynomial for the lateral-directional system matrix, they are

B = −Yβ

V0−Nr′− Lp′ ,

C =Nβ′ + Lp′

Yβ

V0+Nr′

� �+YβNr′V0

−Np′Lr′,

D =Np′YβLr′V0

+ Lβ′ !

− Lp′Nr′Yβ

V0+Nβ

′ !

−Lβ′gV0

,

E = Nr′Lβ′ −Nβ′Lr′

� � gV0

:

ð5Þ

First to consider first-order spiral mode. The time con-stant λS is customary assumed to be jλSj≪min ðωDR, jλRjÞ.From that,

D ≅ ω2DRλR: ð6Þ

2 International Journal of Aerospace Engineering

Consider the classic roll approximation which is given by

λR = −Lp′ ð7Þ

that is found by considering a single DOF roll subsidencemodel [20]. Accordingly, as long as the relationship jλSj≪min ðωDR, jλRjÞ can be satisfied, from equations (5), (6),and (7), the classic Dutch roll natural frequency approxima-tion yields

ω2DR =Nβ

′ +YβNr′V0

−Np′Lp′

Lβ′ +YβLr′V0

!+Lβ′Lp′

gV0

: ð8Þ

In order to demonstrate that weathercock stability is notessential to Dutch roll stability and investigate the crucialfactor, we move the focus on the right terms of equation(8) for the result. As can be seen in the right polynomial,the first three terms are positive. For most of the aircraftconfigurations and flight conditions, weathercock stabilityaerodynamic coefficient Nβ

′ is relatively big among the rightfour terms, where the third term on the right side is usuallyvery small. Therefore, when simply setting Nβ

′ to zero, thesecond term on the right always remains positive to guaran-tee the residual Dutch roll stability, and this can be achievedby a relatively big Yβ.

The vertical tail has been widely recognized as essentialfor most fixed-wing aircrafts to achieve lateral-directionalstability, particularly in the Dutch roll mode stability. Fromthe perspective of reducing sideslip and keeping steady, theyawing moment and side force produced by vertical tail bothmake effort. The related two aerodynamic coefficients are Nβ

′and Yβ. This amounts to saying that although the weather-cock stability can affect the Dutch roll mode, it is not essentialfor the Dutch roll stability, and the side-force derivative Yβ

should be another crucial factor. In fact, with respect to theflight quality specifications, zero weathercock stability butwith sufficient Nβ

′, the lateral-directional mode characteris-tics can satisfy the flight quality specification.

3. Flight Experiment Verification with ZeroWeathercock Stability

To verify the algebraic approximation result, a model gliderwith the conventional configuration was selected as the flighttest object with reduced weathercock stability but maintainedside force stability. The glider has a wingspan of 1.68m, alength of 0.87m, and a mass of 0.714 kg, and it is poweredby an electric motor-driven propeller. The inertia momentparameters relative to lateral-directional motions can befound in Table 1. To have zero Nβ

′ but a marginal effect onYβ, the aircraft was modified by reducing the area of the ver-tical stabilizer to less than one-fourth and attaching a fin thatwas almost three times larger in front of the center of gravity.

The original and modified aircrafts are shown inFigures 1 and 2. The typical level flight condition for thisglider is 18m/s at 100m in altitude.

The aerodynamic parameters were obtained from a vor-tex lattice method program [21]. The vortex layout of themodified configuration is shown in Figure 3. In Table 2, sev-eral representative aerodynamic derivatives involving lateral-directional stability are listed to show the difference betweenoriginal and modified configurations. Then, three groups ofeigenvalues of system matrix Alat were obtained by substitut-ing the aerodynamic derivatives and inertial parameters.Here, the standard primed derivatives Lβ′ and Nβ

′ as well asother relevant derivatives are given by [20]

Li′=Li + Ixz/Izð ÞNi

Ix − I2xz/Iz,

Ni′=Ni + Ixz/Ixð ÞLiIz − I2xz/Ix

,

Yβ =Cyβq ∗ S

mV0,

Lβ = Clβq ∗ Sb,Lp = Clpq ∗ Sb,

Lr = Clrq ∗ Sbb

2V0,

Nβ = Cnβq ∗ Sb,

Np = Cnpq ∗ Sbb

2V0,

Nr = Cnrq ∗ Sbb

2V0:

ð9Þ

The calculated three groups of lateral-directional modeeigenvalues and the corresponding parameters are shown inTable 3. According to the Level 1 criterion in MIL-F-8785C, the time constant of roll mode τR should be less than1.4 s, and the time τdouble to double the amplitude of the spiralmode to 20 degrees should be longer than 20 s. For the Dutchroll mode, its damping ratio ζDR , natural frequency ωDR, andtheir product ζDRωDR should be larger than 0.08, 0.4, and0.15, respectively. All three lateral-directional mode charac-teristics of the modified configuration including Dutch rollmode satisfy the Level 1 specification.

As can be seen in Table 3, the Dutch roll mode frequencyof modified configuration is smaller compared with that oforiginal configuration. This can be attributed to the signifi-cantly reduced Cnβ. From equation (8), the reduction of

Cnβ is equivalent to the decrease of Nβ′ on the right side,

Table 1: Inertia moments involving the lateral-directional stabilityfor the modified configuration.

Parameters Values kg ⋅m2� �

Ix 0.046

Iz 0.074

Ixz 0.001

3International Journal of Aerospace Engineering

thence caused the oscillation frequency ωDR decrease. Thetrend of frequency between these configurations is reason-able from this perspective.

Before the flight test, a lateral-directional motion simula-tion of the modified configuration is made and β and ϕ after a10° sideslip disturbance are shown in Figure 4.

The glider with the modified configuration was flight-tested by remote control with no stability augmentation.Several rudder doublet excitations were conducted, andthe convergence of the Dutch roll motion demonstratesthe consistency between the flight test and the numerical

result. A representative sequential shooting of the aircraftresponse to the rudder doublet is shown in Figure 5. Thisconvergence indicates that the aircraft can achieve Dutch rollmode stability without the weathercock stability.

Two noticeable outcomes besides appeared in the flighttest according to the controller’s feedback. Firstly, themodified glider requires additional rudder control inputin turning compared with the original configuration. Forconventional aircrafts, sideslip arises but within a limit at arelatively small value with no rudder control conducted inturning maneuvers. But for the modified glider, obvious side-slip was observed when conducting a turning maneuver ifrudder control was not involved, and the nose heading didnot change much until rudder control input is towards thesame direction. This can be explained as no yawing momentto reduce the sideslip because of the lack of weathercock sta-bility. The simulation of the sideslip and rolling angle and therolling and yawing angular velocity after the glider encountera 10° right rolling angle disturbance is shown in Figure 6 tomake a comparison between original and modified gliderson this issue. As can be seen in Figure 6, for the original air-craft, the 10° rolling angle soon results in a relatively big yaw-ing angle velocity up to 15°/s and goes on increasing to thesame direction but the sideslip angle remains under 2°. Onthe contrary, the modified one is in big sideslip angle up to8 at the beginning but yawing angular velocity increasesslowly and remains under 10°/s.

Secondly, the modified glider was easier to enter spinsthan the original configuration. The flat spin occurred twicein climbing soon after the aircraft took off. During the climb-ing and turn at a slow airspeed, the sideslip of the glider was

Figure 1: 3-dimensional view of the original configuration.

188 mm 568 mm

Figure 2: Contrast between the modified and originalconfigurations from the side view.

Figure 3: The aerodynamic calculation vortex lattice layout.

Table 2: Stability derivatives of lateral-directional modecharacteristics for the original and modified glider configurations.

Lateral-directionalstability derivative

Originalconfiguration

Modifiedconfiguration

Cnβ 0.0593 0.0001

Cyβ -0.1835 -0.1807

Cnr -0.0477 -0.0188

Table 3: Lateral-directional mode characteristics of the modifiedand original configurations.

Dutch roll mode

Configuration λDR ζDR ζDRωDR ωDR

Modified −0:3772 ± 1:535i 0.239 0.377 1.581

Original −0:773 ± 3:748i 0.202 0.773 3.826

Spiral mode Roll mode

Configuration λS τdouble λR τR

Modified -0.0368 N/A -1.959 0.0557 s

Original 0.181 3.823 -18.120 0.0552 s

4 International Journal of Aerospace Engineering

−5

−40 1

𝛽

2 3 4 5t (s)

6 7 8 9 10

−2

0

2

0𝛽 (°

)𝜙

(°)

5

10

Ф

Figure 4: β and ϕ of the modified glider after 10° sideslip disturbance in lateral-directional simulation.

Rudder doublet input

Figure 5: Configuration modified aircraft response to the rudder doublet Dutch roll mode excitation.

02𝛽

(°/s

)p

(°/s

)

468

0

5

−4−2

0

0

Original

1 2 3t (s)

4 5

24

6

10152025

5r (°/s

)Ф

(°/s

)

101520

Modified

Figure 6: Lateral-directional motions of modified and original gliders after 10° rolling angle disturbance.

5International Journal of Aerospace Engineering

difficult to realize and eliminate in time. Thus, with theincrease in sideslip angle and big attack angle in climbing atlow airspeed, one of the wings stalled; then, the glider entereda spin and quickly developed to a flat spin because there wasno weathercock stability to reduce sideslip, as shown inFigure 7. From the sequential photos, obvious left sideslipduring the right flat spin can be observed and this was diffi-cult to recover under conventional spin recover operation,that is, same aileron and opposite rudder. The reduced rud-der should cause the result.

4. Discussion on the Phenomena andRelevant Inspirations

Both algebraic analysis and flight experiment illustratethat the weathercock stability is not essential to the Dutch rollmode stability. However, in flight appeared the phenom-ena that additional rudder control requirement in turningmaneuvers and easy occurrence of spins reflect the importantroles the weathercock stability plays in flight. The conver-gence of the Dutch roll mode demonstrates that, withinthe flight quality judgment criterion based on the lateral-directional mode characteristic specifications defined inMIL-F-8785C or MIL-STD-1797A, the Dutch roll dynamicstability can be achieved with zero weathercock stability.But those two phenomena can bring bad effect in flight andthey cannot be avoided by just satisfying the mode criterionsin MIL-8785C and MIL-STD-1797A.

In turning maneuvers, the additional rudder controlrequirement of the modified glider reflects that Cnβ playsa more important role in limiting the sideslip angle duringthe maneuvers rather than in the Dutch roll mode stabil-ity. In addition, with less or even zero weathercock stabil-ity, the characteristics of the rudder control requirement incoordinated turns should also be investigated and set areference standard.

Meanwhile, the lack of weathercock stability makes itmore difficult to recover from the beginning of spin. Suf-ficient weathercock stability can restrict the sideslip from

going serious when spin happens; thus, the aerodynamicstate on wings will be improved. As can be inferred thiswill help to prevent the spin from developing and recoverto normal flight.

While neither MIL-F-8785C nor MIL-STD-1797Ainvolves such relevant specifications on weathercock stabil-ity, the phenomena found in flight experiment cannot bedemonstrated basically by satisfying the mode criterionsaccording to flight quality specification. From the view ofthe aircraft design, the sufficient weathercock stability hasits meanings, regardless by the means of aerodynamics orflight control system. Besides the way like arranging appro-priate vertical stabilizer, the split spoilers on the wing tipscontrolled by flight control system according to the simulta-neously detected sideslip angle can also make the aircraftpossess sufficient weathercock stability when needed. Themethod from the view of flight control system has its specialsignification for flying wing configuration and this is worthfurther detailed research.

The impact of gust wind on flight has been investigatedfrom different aspects like overload control. For the impactof lateral wind gust, it is closely related to the residuallateral-directional stability characteristics of the aircraft. Toinvestigate the influence of zeroed Cnβ on the response of air-craft to the lateral gust wind, the numerical method was used.The simulation is based on the lateral-directional lineariza-tion model of the original and modified gliders. The lateralgust wind was simplified to a disturbance in sideslip angleat 10°. By comparing the motions after the sideslip angle dis-turbance between different models, the influence of Cnβ onthe response to lateral gust wind can be studied. The four typ-ical lateral-directional motion parameters of the responsemotions are shown in Figure 8.

From Figure 8, it apparently shows that the modifiedglider with zeroed Cnβ has less directional and rolling angu-lar rate r and p response after the 10° sideslip disturbancewith zero control input compared with those of the originalglider. Both configurations have sideslip angle convergence,though the modified glider expresses lower oscillation fre-quency which is consistent with the previous numerical

Figure 7: Flat spin in the takeoff and climb phase.

6 International Journal of Aerospace Engineering

and experimental study. From this perspective, the zeroedCnβ can help to reduce the impact of lateral gust wind espe-cially on directional motions. For flying wing and other air-craft configurations with less weathercock stability, they canbe less affected by the impact of lateral gust wind comparedwith bigger weathercock stability configurations.

5. Conclusion

In this paper, the algebraic analysis and flight experiment areused to illustrate that the weathercock stability is unessentialto the Dutch roll mode stability but plays some roles in nor-mal flight. First, an algebraic approach was developed toapproximate Dutch roll mode natural frequency and the lit-eral expression demonstrates that the side force derivativeCyβ can also guarantee the residual Dutch roll stability atzeroed Cnβ. Then, a flight experiment was organized basedon a model glider, which was modified to have zero Cnβ butwith a marginal change on Cyβ. The flight test showed theconvergence of the Dutch roll after procedural mode excita-tion and some other phenomena that are thought as a resultof lack of weathercock stability. The phenomena include theextra requirement of rudder input in the turning maneuverscompared with original configuration and easy occurrence ofspin. The relationship between weathercock stability and theimpact of lateral gust wind is also investigated based on thesimulation of the two glider configurations. The result showsthat with less weathercock stability the impact of lateral gustwind can be weakened.

The convergence of Dutch roll in the flight experimentshowed the consistence with the algebraic and numericalanalysis. With the lateral-directional mode criterions in spec-ifications like MIL-F-8785C, they can be satisfied without theweathercock stability. From the view of flight experiment, theside force derivative Cyβ also plays an important role inDutch roll mode stability.

Even though a residual Dutch roll mode stability can beachieved with zero Cnβ, the weathercock stability still showsits roles played in flight. The extra rudder control input inturning maneuvers and easier occurrence of spin demon-strated its effects. These problems cannot be demonstratedor avoided by merely satisfaction of mode criterion in flightquality specifications. Hence, the appropriate weathercockstability has its meanings no matter by means of aerodynam-ics or directional stability augmentation. Moreover, the refer-ence criterion of weathercock stability in flight quality isworth further investigation with the development of flyingwing and other unconventional aircraft design.

Nomenclature

Alat: Matrix of system stability derivative in lateral-directional linearization small disturbance equations

b: SpanClβ: Coefficient of rolling moment due to sideslipClp: Rolling damping moment coefficientClr : Coefficient of rolling moment due to the yawing

angular rate

−5

−20

−3

−80 1 2 3 4 5 6

t (s)7 8 9 10

−4048

0369

−100

10

r (°/s

)p

(°/s

)Ф

(°)

203040

0𝛽 (°

) 5

10

OriginalModified

Figure 8: Lateral-directional motions of modified and original gliders after 10° sideslip angle disturbance.

7International Journal of Aerospace Engineering

Cnβ: Coefficient of yawing moment due to sideslipCnp: Coefficient of yawing moment due to the rolling

angular rateCnr : Yawing damping moment coefficientCyβ: Coefficient of side force due to sideslipIx: Inertial moment of the X componentIz : Inertial moment of the Z componentIxz : Inertial product of the XZ componentLβ: Rolling moment derivative due to sideslip angleLp: Rolling moment derivative due to rolling angular

velocityLr : Rolling moment derivative due to yawing angular

velocitym: MassNβ: Yawing moment derivative due to sideslip angleNp: Yawing moment derivative due to rolling angular

velocityNr : Yawing moment derivative due to yawing angular

velocityq ∗: Dynamic air pressureS: Reference wing areaV0: SpeedζDR: Dutch roll mode damping ratioλDR: Dutch roll mode eigenvalueλR: Roll mode eigenvalueλS: Spiral mode eigenvalueτdouble: Time to double amplitudeτR: Roll mode time constantωDR: Natural frequency of Dutch rollYβ: Side force derivative due to sideslip angle.

Data Availability

The aerodynamic coefficient data used to support the find-ings of this study are available from the corresponding authorupon request.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

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