longitudinal stability and dynamic motions of a small passenger wig craft.pdf

7/29/2019 Longitudinal stability and dynamic motions of a small passenger WIG craft.pdf

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Ocean Engineering 29 (2002) 11451162www.elsevier.com/locate/oceaneng

Longitudinal stability and dynamic motions of asmall passenger WIG craft

H.H. Chun *, C.H. Chang 1

Department of Naval Architecture & Ocean Engineering, Pusan National University, 30 Changjeon-

dong, Kumjeong-ku, Pusan 609-735, South Korea

Received 31 August 2001; accepted 26 September 2001

Abstract

The longitudinal stability characteristics of a Wing-In-Ground (WIG) effect craft are quitedifferent from those of the conventional airplane due to the existence of force and moment

derivatives with regard to height. These stability characteristics play an important role indesigning a safe and efficient WIG due to its potential danger in sea surface proximity. Thestatic and dynamic stability criteria are derived from the motion equations of WIG in theframework of small disturbance theory and discussed in this paper. The static and dynamicstability analyses of a 20-passenger WIG are conducted based on wind tunnel test data, anddynamic motion behaviors are investigated for changes in design parameters. Finally, the flyingquality of the 20-passenger WIG is analyzed at cruising conditions according to the militaryregulations. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Wing-In-Ground (WIG) effect craft; Longitudinal stability; Stability derivatives; Flying quality

1. Introduction

A Wing-In-Ground Effect Craft (WIG hereafter) which flies in a very high speedrange near the sea surface has recently been paid much attention worldwide for futuresuper high speed marine craft use. Because of its increased lift-drag ratio due to theground (or sea surface) effect, the WIG may be economical compared with aircraft.Rozhdestvensky (1996) reported that based on the Russian WIG data, WIG can reach

* Corresponding author. Tel.: +82-51-510-2341; fax: +82-51-512-8836.

E-mail address: [email protected] (H.H. Chun).1 Present address: Samsung Ship Yard, South Korea.

0029-8018/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 2 9 - 8 0 1 8 ( 0 1 ) 0 0 0 9 8 - 1


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1146 H.H. Chun, C.H. Chang / Ocean Engineering 29 (2002) 11451162

Nomenclature

A, B, C, D, E, F Coefficients of the characteristic equationalt Lift curve slope of the tail wing

c Reference wing chord (main wing)

CD Drag coefficientCL Lift coefficientCm Pitching moment coefficientg Acceleration due to gravityh Height of model above ground plane (measured to trailing edge of

tip wing)

h/c Ground clearance nondimensionlized by the reference chord length

it Tail wing incidence

Iy Inertia moment about the y-axis

L, M, N Components of resultant moment about the x, y, z axes

lt Length from C.G. to aerodynamic center of the tail wing

m Mass of WIGna

Ratio of steady-state normal acceleration factor change to angle of

attack change

p, q, r Angular velocity about the x, y, z axes

T Thrust

Tr Reference of momentu, v, w Velocity along the x, y, z axes

Ue Forward velocity at equilibrium

VT Tail volume

X, Y, Z Components of resultant aerodynamic force about the x, y, z axes

Greeks

a Angle of attack

e Downwash angle at the tail wingz Damping ratioq Pitch anglen Root of the speed subsidence modew Frequency

Subscripts

h Differentiation with respect to the dimensionless variable, h/c

u, q, w, a, w Differentiation with respect to the corresponding variable


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1147H.H. Chun, C.H. Chang / Ocean Engineering 29 (2002) 11451162

and exceed the liftdrag ratio of aircraft in spite of relatively small aspect ratios.Unlike aircraft, as a WIG even with a fixed angle of attack approaches the ground, itsforce and moment vary due to the ground effect. Therefore, the longitudinal stability

characteristics of the WIG are quite different from that of conventional aircraft dueto the existence of force and moment derivatives with regard to height. These stability

characteristics play an important role in designing a safe and efficient WIG due toits potential danger in sea surface proximity. Kumar (1969), Irodov (1970), Staufen-

biel (1987) and Hall (1994), conducted studies on the stability of WIG and recently,

Delhaye (1997) reported that by comparing the motion equations of Irodov, Staufen-biel and Hall, these three equations are fundamentally the same as one another.

Stability is a very important factor in the design of a ship and also an airplane.

A lack of stability in the craft could lead to a serious accident and damage. Craft

with an excessive stability (margin) may, however, be insensitive to active control.

In general, the correlation between the longitudinal and lateral motions of the airplane

is weak and the modes of the two motions can be treated separately. It is known

that the WIG inherently possesses lateral stability since the lower wing side of a

banked WIG is subject to increased lift and accordingly, resulting in the restoring

moment. Therefore, most of the stability research of the WIG has been devoted to

the longitudinal problem and this paper is also concerned with longitudinal stability.

The aerodynamic derivatives (lift, drag and moment) of the WIG vary with height,

and their behaviors are strongly non-linear. Therefore, the stability characteristics of

the WIG may be quite different from those of an airplane. In order to investigate

how the WIG is stable, the static stability, which considers only the moment bal-ance by neglecting the inertia and time dependent terms, can first be evaluated, andthen the dynamic stability considering the inertia and time dependent terms can sub-

sequently be evaluated.

In this paper, static and dynamic stability conditions are derived from the longi-

tudinal motion equations of the WIG. The sea surface variation is neglected, and the

sea surface is treated as a rigid wall; thus the sea surface effect can be called theground effect (as it will be referred to hereafter). Based on comprehensive wind

tunnel test results for a 20 passenger WIG, its static and dynamic stability character-

istics are analyzed, and also the dynamic motion behaviors are investigated for vari-

ations in design parameters such as cruising height, cruising speed and the momentof inertia. Finally, the flying quality of the 20-passenger WIG is analyzed at thecruising condition according to the military regulations.

2. Longitudinal static stability

2.1. Pitch stability

First, it is necessary that a WIG should be stable in pitch like an airplane. Anairplane is stable if, after a disturbance in pitch (usually a gust), it returns to the

undisturbed position. The airplane is statically stable if the resultant moment about


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the C.G. (center of gravity) decreases the angle of attack, mathematically given as

follows:

Cma 0 (1)

where Cm is the moment coefficient, a is angle of attack and subscript means thedifferential. The notations and symbols used in this paper are all given in the

nomenclature and if necessary, explanations are added in the text.

2.2. Height stability

Unlike an airplane, the force and moment of a WIG varies with the change of

height. Therefore, an additional static balance condition should be considered.

Namely, a WIG is stable if, after a disturbance in height, it returns to the undisturbedcondition. This can be mathematically expressed as follows, given by Irodov (1970)

and Staufenbiel (1987):

H.S. (Height Stability) CLz 0 (2)

where z is the vertical axis (positive is upwards) and CL the lift coefficient. Sincethe force and moment change with height for the WIG, the following derivatives

can be considered.

dCL CLada CLzdz

dCm Cmada Cmzdz (3)

After removing da in Eq. (3) and rearranging the above equations, the followingheight stability equation can be derived.

H.S. Cma

CLa

Cmz

CLz 0

or,

H.S. XaXz 0 (4)

where Xa

and Xz are the aerodynamic centers of pitch and of height, respectively,

and the derivatives are with respect to the leading edge. This equation can be inter-

preted as that, in order to secure the WIG to be statically stable, the aerodynamic

center of height is located upstream of the aerodynamic center of pitch.

3. Longitudinal dynamic stability

As shown in Fig. 1, a body fixed Oxyz frame is used and O is at the C.G. The

positive z is vertically downwards and Oy is in the starboard direction.The longitudinal linear equations of motion for an airplane with the fixed stick

are given as follows:


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Fig. 1. Axes, force and velocity components.

muXww Xuu Xwwmgq Xqq

mwZww Zuu Zww (Zq mUe)q

IyqMww Muu Mww Mqq (5)

The overdot denotes the time derivative. Kumar (1969), Irodov (1970) and Stau-

fenbiel (1987), introduced the following height derivatives for the WIG in the

above equation:

Xh X

h, Zh

Z

h, Mh

M

h(6)

Then, Eq. (5) can be written as follows:

muXww Xuu Xwwmgq Xqq Xhh

mwZww Zuu Zww (Zq mUe)q Zhh

IyqMww Muu Mww Mqq Mhh (7)

In addition, the following kinematic condition is added:

h w Ueq (8)

Eqs. (7) and (8) can be written in state space form and can be written in matrix

form as follows:

Mx Ax (9)

where M is the mass matrix, A the state matrix, x the state vector (namely

(u,w,q,q,h)T

). In order to investigate the longitudinal stability characteristics of theWIG, the characteristic equation of the system needs to be evaluated. This can be

done by taking the Laplace transform of Eq. (9), assuming the zero initial condition,


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and then by calculating the determinant of (sIM1A), where I is the unit matrix

and s is the Laplace variable. Whereas an airplane out of the ground effect is afourth order system, a WIG craft in ground effect is a fifth order system. Therefore,

the characteristic equation is a fifth order form:

As5 Bs4 Cs3 Ds2 Es F 0 (10)

The coefficients A, B, C, D, E and F are given in Appendix A. The dynamicstability of the system can be evaluated by the RouthHurwitz criterion given asfollows (see Delhaye, 1997; Gera, 1995):

B 0

BCD 0

D(BCD)B(BEF) 0

D(BCD)(BEF)B(BEF)2F(BCD)2 0

F 0 (11)

If the above criteria are satisfied, there is no positive real part in the roots of thecharacteristic Eq. (10) and then the system is stable. If the speed variation terms are

neglected in the last inequality in Eq. (11), this inequality leads to the static height

stability criteria as given in Eq. (4). This means that the dynamic stability cannot

be met without the static H.S. Therefore, when a WIG is designed, it is important

to satisfy the static H.S. first and then consider the dynamic stability.The characteristic Eq. (10) has five roots which can be classified into two oscillat-ing modes which are Short Period Pitching Oscillation (SPPO) and the Phugoid, and

a first order subsidence mode. Then, the characteristic equation can be written inthe following form:

A(s2 2zspwnsps w2nsp)(s

2 2zphwnphs w

2nph)(s n) 0 (12)

where w is the frequency and z the damping ratio, and the subscripts sp and phstand for SPPO and Phugoid, respectively.

4. Stability analysis of a 20 passenger WIG

Shin et al. (1997) designed a 20 passenger WIG, and its aerodynamic character-

istics together with some wind tunnel test results were published. Comprehensive

wind tunnel tests with this craft were conducted and reported by Chun (1997). Table

1 shows the principal dimensions of this WIG, and the model tested in the wind

tunnel is shown in Fig. 2. The aspect ratio of the main wing of this craft is 0.9

which is relatively small; this WIG was designed to be operated in relatively smooth

waters, namely, a maximum cruising height of 0.8 m (equivalent to 0.08 c) with an

operating speed of 150 km/h. The endplates (or sidedplates) are attached to the tipof the main wings as seen in Fig. 2. It is known that an S-shape for the main wing

cross section is good in view of stability, particularly in close proximity to the


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Table 1

Main particulars of the 20-passenger WIG

Length overall 17.45 mBreadth overall 10.60 m

Height overall 5.42 m

Breadth overall 2.20 m

Incidence angle of main wing 3.5 deg

Incidence angle of tail wing 8 deg

Weight 7.5 ton

Mean aerodynamic chord (c) 10 m

Max cruising height 0.08 c (0.8 m)

Cruising speed 150 km/h

Fig. 2. Model of the 20-passenger WIG tested in a wind tunnel.

ground; thus, the main wing cross section is an S-shape. A detailed description on

the craft together with experimental wind tunnel test results can be found in Chun

(1997). Based on these experimental results, a stability analysis of the 20 passengercraft is carried out.

4.1. Static stability

The static stability of the craft is evaluated and its result is shown in Table 2.

According to Table 2, the craft is statically stable even at the limit of the cruising

height of 0.08 c. This means that the craft is automatically stable below this cruising

height. It can be seen that the craft becomes unstable at 0.1 c. As the height decreases,

the aerodynamic center of the pitch moves backwards. The reason for this is that as

the craft approaches the ground, the variation of Cma is relatively larger than thatofCLa. However, as the height decreases, the aerodynamic center of the height moves

considerably forwards. This is due to the fact that since the cross-section of the main


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Table 2

Analysis of longitudinal static stability of the 20-passenger WIGa

C.G. position (Tr) 0.3c

h / c 0.08 h / c 0.1

Cma 0.696144 0.468565

CLa 4.38886 4.21926

Cmz 0.34341 0.256413

CLz 5.03373 2.22069

Xa TrCma/CLa 0.458616 c 0.411054 c

Xz TrCmz/CLz 0.368222 c 0.415466 c

H.S. 0.09 c (stable) 0.004 c (unstable)

a The derivatives are calculated at C.G. and are from the leading edge.

wing of the craft is S-shape, and its moment variation is known to be relatively

insensitive to height change compared with that of a normal wing section, the

increase of Cmz is small and the increase of CLz is very rapid in proximity to the

ground. In addition, the rapid increase of CLz is partly due to the fact that the model

has a lower aspect ratio with the endplates. It is known from the experimental results

of Chun et al. (1996) that an increase of CLz for a wing with a lower aspect ratio

with endplates in proximity to the ground is much larger than that for a higher aspectratio wing.

4.2. Dynamic stability

4.2.1. Evaluation of stability derivatives

The stability derivatives of the 20 passenger WIG are evaluated from the windtunnel tests and shown in Table 3. All the physical quantities are derived from the

experimental data, but the lift curve slope of the tail wing w.r.t. a(alt), downwashangle e, and the downwash slope angle w.r.t. a(e

a) are evaluated by the method in

Roskam (1979).

4.2.1.1. e, ea and alt evaluation Since the downwash is not uniform along the spanof the tail wing, it is usually taken as the mean value over the tail wing for the

stability derivatives. The velocity components over the tail wing can be measured

by various methods (say, pitot tube, LDV etc.) for the model without the tail wing

at different angles of attack. However, these components would be more or less

different from those with the tail wing and it is not easy to obtain the exact velocity

components on the tail surface although these approximate values can be obtained

from much effort in the experimental works. Instead, if there is a verified compu-

tational method that can well predict the aerodynamic coefficients, this can be effec-tively used.

Fig. 3 shows the lift coefficient variation calculated by VLM vs the angle of attack


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Table 3

Stability derivatives of 20 passenger WIG

Stability derivatives UK-style expressions h/c

0.08 0.1

Xu 0.0882 0.08242CDUe

CD

u

1

0.5rSUe

T

uXw CLCDa 0.118304 0.078376

Xq Negligible

Xh 0.126232 0.07503CD/

h

c

Xw Negligible

Zu 0.9826 0.81682CLUe

CL

uZw CLaCD 4.43296 4.26046

Zq VTalt 0.745601 0.74453

Zh 5.03373 2.22069CL/

h

cZw VTaltea 0.132103 0.14087Mu Negligible

Mw Cma 0.696144 0.46856

Mq VTaltlt/ c 0.585297 0.58446

Mh 0.34341 0.2564Cm/

h

c

Mw VTaltealt/ c 0.103701 0.11058

Fig. 3. CL vs angle of attack with different it at h/c 0.08.


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Fig. 4. e vs angle of attack at h/c 0.08.

at h/ c 0.08 with and without the tail wing. The VLM code was verified to shownumerical results that agree well with experimental data for various wings in ground

effect and also WIG (see Chung et al., 1998). In the figure, the tail wing angle ofattack it is varied and also the lift coefficient without the tail wing is included. Sincethe tail wing section is symmetric, the lift should be zero for the effective tail wing

angle of attack being zero. Therefore, the body angle of attack (a) at the cross pointsof a, b, c and d which meet with the lift curve of the craft without the tail wing is

read from the figure, and the mean downwash angle can be obtained by adding itas follows:

e a it

This downwash angle is drawn vs the angle of attack in Fig. 4 from which ea

canbe derived.

Similarly, e and ea at h/ c 0.1 can be evaluated, and are given in Table 4 togetherwith e and e

aat h/ c 0.08. Since the cruising height of the WIG is very low and

the vertical position of the tail wing is relatively high, it can be understood that eand e

aare relatively small. As the craft approaches the ground, alt increases very

little, and e and ea are decreased a bit.By substituting the stability derivatives given in Table 4 into Eq. (10), the eigenva-lues of the characteristic equation for the craft can be obtained. The dynamic per-

turbed motion behaviors of the craft due to cruising height, cruising speed and longi-

tudinal moment of inertia are investigated.

Table 4

Downwash angle (e), ea

and alt with variations of height

h/c e ea

alt

0.08 0.664837 0.177177 4.19863

0.1 0.695228 0.189203 4.19262


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4.2.2. Cruising height changes

The eigenvalues of the system for 150 km/h and Iy 72,456 kg/m2 at two cruising

heights of h/c 0.08 and 0.1 are given in Table 5 and the time responses are shown

in Fig. 5. The perturbed quantity is the dimensionless pitch motion.It can be seen that there is a relatively smooth motion change in SPPO due to the

cruising height variations. As the cruising height decreases, wnsp increases but zspdecreases. It is generally known that wnsp in the SPPO Mode is much influenced byCma. As the craft approaches the ground, Cma increases rapidly, resulting in an

increase in wnsp. However, a dramatic change in Phugoid can be seen at the twocruising heights. At h/c 0.08, the motion is stable but it becomes unstable at

h / c 0.1. This is due to the fact that as shown in Table 2, the static stability con-

dition cannot be satisfied at this height. As the cruising height decreases, the dampingratio zph increases. This can be explained by the fact that in the ground proximity,the air trapped between the ground and the underside of the wing with the endplates

acts as a spring (or air cushion), resulting in an increased damping in the Phugoid

mode.

4.2.3. Cruising speed changes

The eigenvalues of the system for h/c 0.08 and Iy 72,456 kg/m2 at three

cruising speeds are given in Table 6 and the time responses are shown in Fig. 6. As

the speed increases at the given cruising height, the damping ratio is almost

unchanged but the frequency is increased for the SPPO mode. However, it can be

seen that for the Phugoid mode, the damping ratio and motion frequency togetherincrease as the cruising speed increases.

4.2.4. Moment of inertia changes

The results for the three different moment of inertias are shown in Table 7 and

Fig. 7 for h/ c 0.08 and cruising speed of 150 km/h. It can be seen that the motion

frequency and damping ratio decrease with increasing the moment of inertia.

Table 5Eigenvalues of the system for different heights

Height Mode Speed subsidence

SPPO Phugoid

0.08 c 1.32134 3.18131i 0.131638 1.4466i 0.0693

wnsp 3.4448 wnph 1.45264zsp 0.383575 zph 0.0906198

0.1 c

1.2115

2.55985i 0.111827

0.309552i

0.743wnsp 2.83206 wnph 0.329132zsp 0.42778 zph 0.339763


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Fig. 5. SPPO (top) and Phugoid (bottom) for two cruising heights.

5. Flying quality analysis

The requirement for wnph is not specified since it is varied by the cruising speedof the aircraft, as seen in the previous section. Since zph influences the longitudinalmotions when the cruising speed is changed and hence affects the pilot comfort, it

should not be too small and the minimum requirement is, in general, given. The

short period pitch motion is influenced by wnsp and zsp. The minimum and maximumvalues for wnsp are simultaneously specified. Since the requirements for the motionfrequency and damping ratio are not explicitly specified in the regulations of the


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Table 6

Eigenvalues of the system for different speeds

Speed Mode Speed subsidence

SPPO Phugoid

150 km/h 1.32134 3.18131i 0.131638 1.4466i 0.06928

wnsp 3.4448 wnph 1.45264zsp 0.383575 zph 0.0906198

200 km/h 1.75815 4.24242i 0.19069 1.9251i 0.06933

wnsp 4.5923 wnph 1.93452zsp 0.382847 zph 0.0985723

250 km/h 2.19556 5.30337i 0.247171 2.40425i 0.07326

wnsp 5.73988 wnph 2.41692zsp 0.38251 zph 0.102267

civil aircraft (FAR 23, FAR 25, JAR-VLA etc.), the design of the civil aircraft, in

general, follows the military regulations.

The 20 passenger WIG can be classified in Class II, Category B (see Roskam,1985; ESDU, 1992), but for the flying quality analysis, it is assumed as Category

A which is more severe than Category B. Again, the craft is a passenger carryingone, so it should be designed to satisfy the flying quality of Level I at the cruisingconditions. Level I requirements for MIL-F-8785C and MIL-STD-1797A are given

in Table 8.

It can be seen from Fig. 8 that the 20 passenger WIG at a cruising height of

h / c 0.08 satisfies the Level I requirements. In addition, the control anticipationfactor (CAP), which is 1.33 for the craft, also satisfies the Level I requirements. Fig.9 shows the typical pilot opinion contours for the short period together with the

value for the present craft. In conclusion, it can be evaluated that the dynamic motion

behaviors of the present 20 passenger craft seem to be good.

6. Conclusions

The longitudinal static and dynamic stability criteria of the WIG are discussed.

Based on the wind tunnel results together with VLM code, the stability derivatives

for the 20 passenger WIG are evaluated and its static and dynamic stability character-

istics are investigated. The dynamic motion behaviors of the craft are also investi-

gated by varying the design parameters such as cruising speed, cruising height and

moment of inertia, and the flying quality is analyzed. It is shown that the craft seems

to be good in terms of stability and dynamic motions.Stability characteristics play an important role in designing a safe and efficient

WIG due to its potential danger in sea surface proximity. One of the easiest ways


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Fig. 6. SPPO (top) and Phugoid (bottom) vs time for three cruising speeds.

in increasing the static stability and the damping ratio for the SPPO is to increase

the tail wing size at the cost of the increased structural weight and the drag increase.

These penalties would deteriorate the merit of the WIG concept. Therefore, it is

important to design a WIG, which satisfies the static stability and also the dynamicmotion characteristics within the flying quality limit as well as maximizing the liftdrag ratio.

Acknowledgements

The authors would like to thank Dr. M.S. Shin, Korea Research Institute of Ship

and Ocean Engineering, for allowing the use of the experimental data.


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Table 7

Eigenvalues of the system for different moments of inertia

Iyy Mode Speed subsidence

SPPO Phugoid

72,456 kg/m2 1.32134 3.18131i 0.131638 1.4466i 0.06928

wnsp 3.4448 wnph 1.45264zsp 0.383575 zph 0.0906198

100,000 kg/m2 1.09512 2.9404i 0.112063 1.35305i 0.06926

wnsp 3.13771 wnph 1.35768zsp 0.349019 zph 0.0825401

120,000 kg/m2 0.998858 2.82969i 0.100562 1.29213i 0.06925

wnsp 3.00081 wnph 1.29603zsp 0.332863 zph 0.0775646

Table 8

Level I requirements for MIL-F-8785C and MIL-STD-1797A

Phugoid damping requirements zph0.04

Short period damping ratio limits 0.35zsp1.30Short period undamped natural frequency

0.28w2nsp

na

3.6a

a Notew2nspn

a

: CAP (Control Anticipation Factor).

Appendix A

A 1

B 1

mIy(mZw)[mMw(mUe Zq) mMq(m Zw)Iy(mXu XwZu

mZwXuZw)]

C1

mIy(mZw)[mMqXu mMwUeXu mIyZh MwXuZqmMw(mUe Zq)

MwXqZuIyXwZu MqXwZu mMqZw IyXuZwMqXuZwMu(mXq mUeXw XwZqXqZw)]

D 1

mIy(mZw)[mMqZhmMwUeZhIyXuZh mMhZq gmMwZu


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Fig. 7. SPPO (top) and Phugoid (bottom) vs time for three moment of inertia values.

IyXhZu MqXwZu Mw(mUeXu XuZqXqZu)MqXuZw mMhUeZw Mu(gm

2mUeXwXwZq XqZwgmZw)]

E1

mIy(mZw)[MqXuZh MwUeXuZhMhXuZqMqXhZuMwUeXhZu

MhXqZu MhUeXwZu Mw(mUeZh gmZu) mMhUeZwMhUeXuZwMu(XqZh UeXwZhXhZq gmZwUeXhZw)]

F 1mIy(mZw)

[gmMuZhgmMhZu Ue(MwXuZhMuXwZhMwXhZu

MhXwZu MuXhZwMhXuZw)]


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Fig. 8. Control anticipation parameter and SPPO damping ratio requirements.

Fig. 9. Typical pilot opinion contours for short period.


18/18


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