1

Stability and Control Analysis of the Zivko Edge 540T

Ariann M Duncan1

Georgia Institute of Technology, Atlanta, Georgia, 30332

The report below is a stability and control analysis of the Zivko Edge 540T. The Edge 540T

is a high performance aerobatic aircraft developed by Zivko Aeronautics Inc. Non-

dimensional longitudinal derivatives were first determined for the aircraft. These were

calculated from a defined equilibrium flight condition of 10,000ft, Mach number of 0.28 and

cruise speed of 303ft/s. Through estimates of these derivatives, it was found that the aircraft

is longitudinally stable. The moment coefficient with respect to the angle of attack Cmα = -1.23

was found to be negative. The non- dimensional lateral-directional derivatives were found

next. It was determined that the aircraft is laterally stable due to the positive value of Cnβ =0.2

Both sets of non-dimensional derivatives for the Zivko Edge 540 were compared to those of a

General Aviation Aircraft. It was found that these two sets of data approximately matched,

indicating reasonable validity of the methods of approximation used. Dimensional derivatives

were next estimated. These were used to determine the natural frequency, damping ratio,

eigenvalues and eigenvectors associated with the phugoid mode, the short period mode, the

spiral mode, the roll mode and the Dutch roll mode. The eigenvalues of the phugoid, short

period, roll and Dutch roll mode were all negative for the real part of eigenvalue. This

indicated that these modes are stable. The spiral mode was the only mode that proved to be

unstable. It is a possibility that this may be accounted for by the performance characteristics

of the aircraft. Behavior of the roll mode with a maximum aileron deflection of 20 degrees

was also investigated and it was proved that the aircraft has a roll rate of 450 degrees per

second at a speed of 303 ft/s. This value approximately matched the roll rate of 420 degrees

per second highlighted by the Red Bull Air Race committee [11].

Nomenclature

CL0 – reference lift coefficient

CD0 –reference drag coefficient

2

W –weight of airplane

S - Area of wing

cr – Root Chord

ct – Tip Chord

c_bar – mean aerodynamic chord

b – Wing Span

Ixx – Roll Moment of Inertia

Iyy- Pitch Moment of Inertia

Izz – Yaw Moment of Inertia

CLα – Lift-curve slope of the airplane

ARwing – Aspect Ratio of the wing

e- Oswald’s efficiency factor

VH – Horizontal tail volumetric ratio

CLαwing Lift curve slope of the wing

St – Area of the horizontal tail

lt – Distance from center of gravity to tail quarter

chord

CLα_hortizontal_tail – Lift-curve slope of horizontal tail

ηtail –efficiency factor of the tail

dε/dα – Change in downwash due to change in AOA

xcg – Center of gravity of the airplane

xac – Aerodynamic center of the airplane

Cmαfus – Pitching moment due to fuselage

dCm/dM – Change in pitching moment with Mach

no.

CDu – Drag due to forward velocity

Sv - Area of the vertical tail

lv -Distance from center of gravity to tail ac

CLαv – Lift-curve slope of the vertical tail

Vv – Vertical tail volumetric ratio

η_vertical tail –Efficiency of the vertical tail

dσ/dβ – Change in sidewash due to sideslip

Cnβwf - Yawing moment due to the wing and fuselage

Λ- Wing Sweep angle

Cyβ_tail –Y derivative with respect to Beta

ARvertical_tail – Aspect Ratio of the vertical tail

T –Taper Ratio

𝜏 – Flap Effectiveness parameter CLu – Coefficient of lift with respect to forward

velocity

Mu - Moment with respect to forward velocity

Mw - Moment with respect to velocity

Mw - Derivative of moment with respect to velocity

Cmα – Coefficient pitching moment with respect to

AOA

3

Cmα - Pitching moment coefficient with derivative of

AOA

Cmu –Pitching moment coefficient with respect to U Cmq – Coefficient of pitching moment with respect to

q – dynamic pressure

Mq –Moment with respect to q

Mα - Moment with respect to AOA

Mα - Moment with respect to derivative AOA

I. Introduction

he objective of the report below is to estimate the stability and control derivatives of the Zivko Edge 540T. The

Edge 540 developed by Zivko Aeronautics is a high performance aerobatic racing aircraft. The aircraft has been

popularized from its use as one of the primary aircraft in the Red Bull Air Race competition [11]. This aircraft was of

particular interest due to its extensive maneuverability in flight. Increased maneuverability can sometimes result in

lack of stability of the modes of the aircraft. This analysis was therefore conducted to determine if the statement above

proved true.

An equilibrium flight condition will first be defined. Following this, aircraft physical properties will be predicted

based on rough estimates from a 3-View of the airplane [13], as well as from data obtained in the Edge 540 owner’s

manual [1]. Non-dimensional and dimensional stability derivatives will then be calculated for longitudinal motion and

lateral motion. The non-dimensional derivatives help to give valid insight into the stability of the aircraft while the

dimensional derivatives help to verify the information obtained from the non-dimensional derivatives. Dimensional

derivatives also provide insight on the specific modes of the aircraft. The stability characteristics of each mode based

on the results of the eigenvectors and eigenvalues of the dimensional derivatives will be highlighted in the section on

“Mode Approximations”. The implications of these modes on the aircraft’s performance will also be discussed.

A summary of the paper referencing key findings will then be outlined and presented to the reader.

II. General Overview of the Zivko Edge 540T

Physical properties and general attributes of the aircraft were estimated from the Pilot’s Operating Handbook and

Flight Manual for the Edge 540T [1]. Dimensions listed on the 3-View drawing in Figure 1 were obtained from the

Fédération Aéronautique Internationale (The World Air Sports Association) [13] and were used to estimate the weight,

T

4

total length and wing span of the airplane. It was found that the values listed in the 3-View drawing approximately

matched those listed in the Pilot’s Operating Handbook and Flight Manual [1].

The reference lift coefficient, CL0 and the reference drag coefficient, CD0 of the airplane were estimated based on

the properties of the airfoil used in the main wing. As the aircraft is aerobatic, it was assumed that it uses a symmetric

airfoil for the wing and horizontal tail. The NACA 0015, a popular airfoil section used on sports airplanes was

therefore used to model the behavior of CL0 and CD0 at an angle of attack of 1 °. This was done using XFOIL at given

flight conditions (See Appendix Figure A9). The mean aerodynamic chord was found using the relationship between

the root chord and the tip chord as shown in Figure A10 of the Appendix. The root chord and tip chord, as well as

other aircraft attributes not listed in the owner’s manual were estimated based on scaling done with the 3-View drawing

and the actual physical parameters of the aircraft, such as aircraft length and wing span.

Table I below highlights key physical parameters of the aircraft’s geometry.

Table I: Physical Properties of the Zivko Edge 540

Physical Property Value

CL0 0.1205

CD0 0.0054

W 1598.35 lb

S 106ft2

cr 5.5 ft

ct 2.75 ft

c 4.27ft

b 25.83 ft

Ixx 2605 slug-ft2

Iyy 1620 slug-ft2

Izz 3605 slug-ft2

5

Figure 1: 3-View Drawing of the Edge 540T [13]

In order to analyze the stability and control derivatives of the aircraft, a specific flight equilibrium

condition was first chosen.

The Edge 540 has a service ceiling of 16,000 feet [1], thus an altitude of 10,000ft was selected for the

equilibrium flight condition. This is below the ceiling but still within the aircraft’s normal range of operation. This

altitude could be used as a cruise condition in setting the airplane up for an aerobatic maneuver. Additionally, the

Edge’s average speed at cruise is approximately 303ft/s [1]. This speed was selected in the analysis in order to

correspond to cruise conditions. The Mach no. was determined using equation 1 where the speed of sound a, was

predicted based on the temperature at an altitude of 10,000ft. The value was found from “Speed of Sound vs.

Altitude” [12].

M = v/a (1)

The density, ρ at the equilibrium condition was also estimated by considering atmospheric conditions at 10,000 ft.

The value was found using “International Standard Atmosphere Conditions”[3] .

6

Equilibrium flight conditions are shown in Table II below.

Table II: Equilibrium Flight Conditions

Flight Condition Value

Altitude 10,000 ft.

Mach No. 0.28

α 1°

ρ 0.001755 slug/ft2

U0 303ft/s

θ0 0

III. Estimate of Non-Dimensional Stability and Control Derivatives

Longitudinal

The non-dimensional stability and control derivatives were estimated using the equations listed in Figure A1 (See

Appendix)

Cxu was calculated using the reference drag coefficient, CD0 and the thrust coefficient, CTU. The term CDU,

the drag due to the forward speed was set to zero as the aircraft is travelling at low (subsonic) speeds [14].

Additionally the aircraft is propeller driven, which implies CTU = - CD0. Thus Cxu is equal to -3CD0.

Cxα was estimated using the values of the reference lift coefficient, the lift-curve slope of the airplane, the

aspect ratio of the main wing and Oswald’s span efficiency factor, e. The lift-curve slope of the airplane

was estimated to be 5.7 (CLα data was unavailable for the Zivko Edge or any other specific aerobatic

aircraft). The value was chosen based on aircraft estimates for CLα in “Flight Stability and Control” by

Nelson [14].

Cmu was estimated based on a Mach no. of 0.28 and the reference lift coefficient.

Czα was found by substituting CLα of the airplane and the reference drag coefficient into the corresponding

equation shown in Figure A1.

Czα was estimated using the efficiency of the horizontal tail, η, the lift curve slope of the tail, CLαt, the

horizontal tail volumetric ratio, VH and the effect due to the downwash, dε/dα. VH was found using

equation 2 where lt was found through scaling done with the 3-View drawing.

VH = (lt*St)/S*c_bar (2)

The downwash was calculated using equation 3

7

dε/dα = 2*CLα_wing /pi*AR_wing (3)

CLα_wing in the equation above was calculated using equation 4 where AR is of the wing.

(4)

Czq was found using values of the tail efficiency factor, the lift curve slope of the tail and the horizontal

tail volumetric ratio. CLα_tail was also found using equation (4) and setting AR to AR of the tail.

In the equation for Czδe, dCLt /dδe was set to a value of 0.5. Values of this parameter could not be found

for the Edge 540T and estimates were chosen based on lower and upper limits of 0 and 1.

Cmu was set to zero as dCm/dM can be neglected at low (subsonic) speeds.

In the equation for Cmα , the center of gravity xcg and the aerodynamic center xac were found through

equations in the Edge 540 owner’s manual. The aerodynamic center was defined at 4.5 feet while the

c.g of the aircraft was calculated by dividing the total moment by the weight of the airplane [1]. The

c.g was found to be located at 3.197 feet. The pitching moment coefficient with respect to angle of

attack of the fuselage was found through comparisons with other propeller driven aircraft as found in

“Empennage Sizing and Aircraft Stability ” by Ryan C. Struett [7]. Equation (5) can be used as

alternative for solving Cmαfus.

[8] (5)

Cmα was estimated using the values of tail efficiency factor, η, CLαt, VH , lt and the chord and the

downwash factor dε/dα

Cmq was estimated using the same parameters as Cmα with the exception of the downwash factor.

In the Cmδe term, dCLt /dδe was chosen as stated above at 0.5

To find the non-dimensional derivatives above, additional assumptions as highlighted below were also made:

The Oswald efficiency factor, e was estimated from the Cessna 172 [4], a propeller driven aircraft of a similar size.

The value was found to be within the range of 0.65 to 0.75. Typical tail efficiency factors for conventional aircraft lie

between 0.85 – 0.97, thus the average of this range 0.9 was chosen for η [5].

8

Table III below is a compilation of the non-dimensional longitudinal stability derivatives

Table III: Non-dimensional Longitudinal Stability Derivatives for Zivko Edge 540

Cx Cz Cm

u -0.01641 -0.230749132

0

α 0.02121 -5.70547 -1.23004

q 0 -2.962829 -7.341156

α 0 -1.29503 -3.208765

αe 0 -0.328837 -0.8147781

The values in the table above were compared with those of the STOL transport aircraft in Nelson as well as to the

General Aviation Airplane [14]. The values of Cmα , Cmα ,Cmq, Cmδe have the same sign and order of magnitude of the

General Aviation Aircraft. The variables listed in the table above also have the same signs as the STOL transport

aircraft, although some differ by order of magnitude. This may be due to the differences in size of the STOL

transport plane and the Zivko Edge 540.

From the values obtained, the aircraft appears to be longitudinally stable in cruise.

Table IV: Comparison of various Non-dimensional Longitudinal Stability Derivatives between the General

Aviation Aircraft and Zivko Edge 540

Cm General Aviation Cm Edge 540

u -

0

α -0.683 -1.23004

q -9.96 -7.341156

α -4.36 -3.208765

αe -0.9223 -0.8147781

Lateral-Directional

The non- dimensional lateral directional stability derivatives were estimated using the equation in Figure A2 (See

Appendix)

Cyβ was estimated using the tail efficiency factor, η, the area of the vertical tail Sv, the area of the

horizontal tail S, the lift curve slope of the vertical tail CLαv and the change in sidewash with the

9

change in sideslip angle. The lift curve slope of tail was estimated using equation 4 above. The change

in sidewash, dσ/dβ was estimated 0.3 from data given in Nelson [14].

Cyp was set to zero as the wing has zero sweep angle

In the equation for Cyr , Cyβ of tail was calculated using equation 6, lv was found from scaling the 3-

View.

Cyβ = -CLαv *ηv *(Sv/S) (6)

In the equation Cydr , T flap effectiveness parameter was set to 0.55 from estimates in Nelson.

In the equation for Cnβ, Cnβwf was estimated at 0.93. This value was taken from estimates in

“Empennage Sizing and Aircraft Stability”[7] by Ryan C Struett.

In Cnp, the value of CL was estimated at 0.19. More accurate estimates could have been obtained by

using equation 7

CL = (CL0 +CLα)* α (7)

In the equation for Cnr, Vv was estimated using equation 8 while lv was determined from scaling the

3-View.

Vv =(lv*Sv)S*b (8)

In the equation for Cnδa, K was estimated using Figure 3.12 in Nelson [14]

Cndr was estimated by taking into account vertical tail volumetric ratio, efficiency factor of the tail,

flap effectiveness parameter and the lift curve slope of the tail

For the equation of Clβ, only the ΔClβ term was considered as the wing has no dihedral. ΔClβ was

estimated using Figure 3.11 of Nelson for maximum ordinals on upper surface. [14]

Clp was found using the equation given in the Appendix for a taper ratio of 0.5.

In the equation for Clr, zv was found through scaling of the 3-View with actual aircraft parameters

The integral term in the equation for Cl δa was found by integrating over the length of the control

surface.

Cldr was found using the values of previously estimated variables.

10

Table V: Table showing Lateral – Directional Stability Derivatives for Zivko Edge 540

Cy Cn Cl

β -0.39732 0.2114966 -0.0002

p 0 -0.02375 -0.791667

r 0.4023 -0.264775 0.075292

δa 0 -0.014305 0.329761

δr 0.186774 -0.110633 0.020969

The values of the non-dimensional lateral stability derivatives of the Edge 540 were compared to the General

Aviation Aircraft in Nelson [14]. The highlighted values in Table VI for the General Aviation Aircraft approximately

correspond to the values obtained for the Edge 540. A notable difference is the positive value of Cnβ obtained for the

Edge 540. This suggest that the Edge 540 is laterally stable.

Table VI: Table showing Lateral – Directional Stability Derivatives for the General Aviation Aircraft

Cy Cn Cl

β -0.5646 -0.071 -0.074

p - -0.0575 -0.410

r - -0.125 0.107

δa - -0.0035 -0.134

δr - -0.072 0.107

IV. Estimate of Dimensional Stability and Control Derivatives

Longitudinal

The dimensional longitudinal derivatives were estimated using equations listed in Figure A3 of the Appendix.

The moments of inertia of the airplane listed in Table I above were obtained from an “aircraft.cfg” file in

Microsoft’s Flight Simulator X [15]. The file obtained provided data for the Extra 300. As the Extra is an aerobatic

aircraft with similar dimensions and performance capabilities, the values were used to model and predict the

performance of the Edge 540. In addition to this, the values were compared to those of the Cessna 172 to ensure

relative accuracy [16]

U0 in the equation was used as the aircraft’s cruise speed, 303ft/s. All other dependent variables were calculated

from the non-dimensional derivatives or approximated as described above in the detailed description for calculating

11

the non-dimensional derivatives. The values in Table VII were obtained by applying the relationships shown on page

149 of Nelson.

Table VII: Table showing Dimensional Longitudinal Stability Derivatives for the Zivko Edge 540

Δu -0.0062115 -0.1189497 0 -32.2

Δw -0.142655 -3.23945 303 0

Δq 0.000239544 -0.0859324 -1.67311 0

Δθ 0 0 1 0

The natural frequency and damping ratio of both the short period and phugoid (long period) modes were obtained

using equations 9, 10, 11 and 12. The Phugoid mode was also estimated for negligible compressibility effects using

the equation in “Flight Stability and Automatic Control” by Nelson [14].

ωp = sqrt((-Zu*g)/U0) (9)

ζp = -Xu/ 2 ωp (10)

ωsp =swrt( (ZαMq/U0) -Mα) (11)

ζsp =- (Mq + Mα + Zα/U0 )/2 ωsp (12)

Table VIII: Natural Frequency, Damping Ratio and Eigenvalues of the phugoid mode for the Edge 540

Phugoid Mode

Natural Frequency, ωp 0.123126124

Damping Ratio, ζp 0.025224161

Eigenvalue, λp -0.0031+0.1216i

-0.0031-0.1216i

Phugoid Mode (Neglecting compressibility effects)

Natural Frequency, ωp 0.150289326

Damping Ratio, ζp 0.03209854

Table IX: Natural Frequency, Damping Ratio and Eigenvalues of the short period mode for the Edge 540

Short Period Mode

Natural Frequency, ωsp 5.608697

Damping Ratio, ζsp 0.4379409

Short Period Mode

Eigenvalue, λSp -2.455+5.03716

-2.455-5.03716

12

The eigenvalue of the real part of the phugoid mode is negative, this implies that the mode is dynamically stable.

The eigenvalue of the real part of the short period mode is also negative which suggests that the short period mode is

also dynamically stable. The eigenvalues were obtained using the “eig” function in MATLAB for a matrix of values

in Table VII. The natural frequencies and damping ratios are positive for both modes and are within an acceptable

range.

Lateral-Directional

The dimensional lateral derivatives were estimated using equations listed in Figure A3 of the Appendix.

Variables needed in the equations listed were calculated in the section on non-dimensional lateral derivatives and

substituted into the equations. The dimensional lateral derivatives shown in Table X were obtained by applying the

relationships shown on page 199 of Nelson.

Table X: Dimensional Lateral derivatives of the Zivko Edge 540

Δ β -0.22559 0 -0.963382 0.160271

Δp -0.016935 -2.857255 0.27174 0

Δr 12.9407924 -0.06194 -0.6905362 0

ΔΦ 0 1 0 0

The table of dimensional lateral derivatives shown above was converted to a matrix and the “eig” function in

MATLAB was used to find the eigenvalues of the Dutch Roll mode, the Spiral mode and the Roll mode. The real part

of the eigenvalue for the Dutch Roll mode is negative as shown in the table below which indicates that this mode of

the aircraft is stable. The damping ratio and natural frequency of the Dutch roll mode are in the range of values

expected. They were calculated using equations (5.47) and (5.48) in Nelson [14].

Table XI: Dutch Roll mode of the Zivko Edge 540 (Eigenvalues, Natural Frequency and Damping ratio)

Dutch Roll Mode

Natural Frequency, ωDR 3.6014

Damping Ratio, ζDR 0.12718

Eigenvalue, λDR -0.4608 +3.5279i

-0.4608 -3.5279i

As shown Table XII below, the eigenvalue for the spiral mode is positive, this implies that the spiral mode is unstable.

In addition to this, the eigenvalue of the roll mode is negative which implies that the roll mode of the aircraft is stable.

13

The unstable spiral mode may be due to the aerobatic nature of the aircraft as unstable modes lead to greater

maneuverability. The sign on the spiral mode is expected as the airplane’s wings have little to no dihedral.

Table XII: Spiral mode of the Zivko Edge 540 (Eigenvalues)

Spiral Mode

Eigenvalue, λSP 0.0103

Table XIII: Roll mode of the Zivko Edge 540 (Eigenvalues)

Roll Mode

Eigenvalue, λR -2.8621

V. Mode Approximations

Phugoid Mode

The phugoid mode was approximated using equation shown in Figure A4 of the Appendix. The eigenvalues of the

phugoid mode shown in the table below approximately match the values obtained using the MATLAB function “eig”

with the matrix of the longitudinal derivatives. The real part of the mode is negative which verifies that the mode is

stable.

Table XIV: Phugoid mode of the Zivko Edge 540 (Eigenvalues)

Phugoid Mode

Eigenvalue, λp -0.0031+ 0.121616i

-0.0031- 0.121616i

Table XV: Magnitude and Phase of the Phugoid mode (Eigenvectors)

Magnitude Phase

Δu 0.024 48.535

Δw 0.992 0

Δq 0.0168 81.09

Δθ 0.00302 -34.215

From Table XV above, it can be seen that the phase of Δu and Δθ are approximately 82 degrees out of phase

(48.535+ 34.215). This is expected as u and θ are usually out of phase by 90 degrees.

14

Short Period Mode

The eigenvalues of the short period mode were estimated using the equation in Figure 5A. It was observed that

these approximations matched those calculated above using MATLAB. The real value of the mode is still negative

which implies that the mode is stable. The short period mode is a good approximation for the overall stability of the

aircraft, thus the aircraft can be said to be stable under equilibrium flight conditions.

Table XVI: Eigenvalues of the short period mode

Short Period Mode

Eigenvalue, λsp -2.45 -5.0398

-2.45+5.0398

Table XVII: Magnitude and Phase of the short period mode (Eigenvectors)

Magnitude Phase

Δu 1 0

Δw 0.005 -175

Δq 0.0004 0

Δθ 0.0036 -91.59

Roll Mode and Rolling Motion

The eigenvalues for the roll mode were approximated using the equation in Figure 6A of the Appendix. This is the

value of Lp from the dimensional derivative table for lateral directional stability. Lp was found to be -2.857 which

precisely matches the value obtained in the section above. The negative value suggests the mode is stable.

Table XVIII: Eigenvalues of the Roll mode

Roll Mode

Eigenvalue, λR -2.857255433

Table XIX: Magnitude of Roll mode (Eigenvectors)

Roll

Δβ -0.007

Δp -0.9439

Δr 0.0165

ΔΦ 0.3298

15

Equation 13 below was used to generate the plot shown in Figure 2 below. This model predicts the roll rate

response for a step input with maximum aileron deflection. Maximum aileron angle for the Edge 540 is 20 degrees [1].

ΔP(t) = -L δa /Lp (1-eLp*t) Δ δa (13)

From the plot, it can be seen that the maximum roll rate reaches 450 degrees in 1 second. This value approximately

matches the roll rate of 420 degrees per second highlighted under the airplane performance characteristics by the Red

Bull Air Race organization. [11] .The difference may be accounted for by the values of rolling moment due to aileron,

Clδa. The graph has a small overshoot at 1.5 seconds and levels off after this time, verifying that the mode is stable.

Figure 2: ΔP vs time for the Zivko Edge 540

Spiral Mode

The spiral mode was approximated from the equation in Figure A7. This approximation did not prove valid as it

differed significantly from the value obtained using the “eig” function in MATLAB. It is possible that this error may

have resulted due to an oversight while calculating one of the variables in the Excel spreadsheet used for calculations.

Table XX: Eigenvalues of the Spiral Mode

Spiral Mode

Eigenvalue, λSP 206.960

16

Table XXI: Magnitude of the Spiral Mode (Eigenvectors)

Spiral

Δβ 0.0059

Δp 0.0102

Δr 0.1082

ΔΦ 0.9941

Dutch Roll Mode

The eigenvalues of the Dutch Roll mode were calculated using the equations in Figure A8. The values

approximately matched those calculated for the Dutch Roll mode in the section above. The real value of the mode is

negative which indicates that the mode is stable.

Table XXII: Eigenvalues of the Dutch Roll Mode

Dutch Roll Mode

Eigenvalue, λSP -0.458065+3.43368i

-0.458065-3.43368i

Table XXIII: Magnitude and Phase of the Dutch Roll Mode (Eigenvectors)

Dutch Roll Dutch Roll

Magnitude Phase

Δβ 0.263 86.225

Δp 0.0613 -56.776

Δr 0.962 0

ΔΦ 0.0172 -154.179

Table XXIV: Stability of the Modes

Modes Stability

Phugoid Stable

Short Period Stable

Roll Mode Stable

Spiral Mode Slightly unstable

Dutch Roll Mode Stable

17

VI. Conclusion and Recommendations

In conclusion, the non-dimensional and dimensional stability derivatives for the Edge 540T were determined. The

non-dimensional derivatives gave insight into the longitudinal and latitudinal-directional static stability of the aircraft.

The values of the non-dimensional derivatives for the Edge 540 were found to be on the same order of magnitude of

those of the General Aviation Aircraft defined by Nelson in “Flight Stability and Automatic Control”. It was

determined that the Edge 540 is longitudinal statically stable due to Cmα = -1.23 and lateral statically stable due to Cnβ

=0.2.

The dimensional lateral and longitudinal derivatives were used to determine the behavior of the modes of the aircraft.

It was found that all the modes, with the exception of the spiral mode were stable. The values obtained for the spiral

mode differed significantly in the value calculated from the equation of the mode approximation, Figure A7 and those

calculated in MATLAB. Thus it is uncertain if the data for this mode is accurate. The most likely value would result

from the MATLAB calculations as this value is closer in order of magnitude to those typically observed for this mode.

If the MATLAB data is valid, it proves the mode is unstable which may be an intentional design due to the high

maneuverability of the airplane. The natural frequencies and damping ratios proved to be in the expected range of

values for stability for all modes calculated.

Finally, the roll rate of the aircraft was determined using equation 13 above. A graph of roll rate vs. time was plotted

and it was found that the airplane has a roll rate of 450 degrees at a maximum aileron deflection of 20 degrees. This

value matches the performance characteristic typically seen for roll rate of the Edge 540 at 303ft/s.

It should be noted that some of the derivatives were obtained through rough approximation of parameters and estimates

based on other aircraft of similar scale. Therefore some values may not be precise when compared to actual Edge 540

performance characteristics. More accurate results can be obtained if variables needed to solve the equations

highlighted in the Appendix, Figures A1- A3 were obtained from actual Edge 540 data.

Appendix

Variables used in Excel spreadsheet calculator.

Physical Property Value

CL0 0.1205 CL of Aircraft 0.19

CD0 0.00547 CD α 0.33

18

W 1598.35 m 49.63819876

S Area 106

cr 5.5

ct 2.75

c_bar 4.27

b 25.83

Ixx 2605

Iyy 1620

Izz 3605

CLα 5.7

ARwing 6.294234906

e 0.7

VH 0.514250365

CLαwing 4.319327235

St 22

lt 10.58

CLα_hortizontal_tail 3.200807653 ARhorizontal_tail 2.969988131

ηtail 0.9

dε/dα 0.437092655

xcg 3.197

xac 4.508

Cmαfus 0.93

dCm/dM 0

CDu 0

Sv 14.83

lv 17

CLαv 2.427283271

Vv 0.092078832

η_vertical tail 0.9

dσ/dβ 0.3

Cnβwf -0.05

λ 0

Cyβ_tail -0.305631602

ARvertical_tail 1.800053196

Taper Ratio 0.5

ΔClβ -0.0002

Zv 2.9

𝜏 0

CLu 0.010250868

Mu 0

Mw -0.091372097

Mw -0.001679183

Cmα -1.23

19

Cmα -3.208

Cmu 0

Cmq -7.341156505

Mq -1.164315936

Zα -981.5527855

Mα -27.68574531

Mα -0.508792521

Q 80.5623

Figure A1: Equations for estimating the non-dimensional longitudinal stability coefficients

20

Figure A2: Equations for estimating the non-dimensional lateral stability derivatives

21

Figure A3: Equations for estimating the dimensional stability derivatives

Figure A4: Equation for estimating the eigenvalue of phugoid mode

Figure A5: Equation for estimating the eigenvalue of the short period mode

22

Figure A6: Equation for estimating the eigenvalue of the roll mode

Figure A7: Equation for estimating the eigenvalue of the spiral mode

Figure A8: Equation for estimating the eigenvalue of the Dutch Roll mode

Figure A9: NACA 0015 at 1 degree AOA

23

Figure A10: Estimating the mean aerodynamic chord

Figure A11: Edge 540

References

[1] “Pilot Operating Handbook and Flight Manual” (Zivko Aeronautics Inc , 2016)

http://www.kfva.ch/fileadmin/images/downloads/Edge-Manual.pdf

[2] “NACA 0015” (Airfoil tools) http://smiller.sbyers.com/temp/AE510_03%20NACA%200015%20Airfoil.pdf

[3] “ISA Conditions” http://www.hyperflite.co.uk/isa-conditions.html

[4] John McIver, “Cessna Skyhawk II/10 Performance Assessment” http://temporal.com.au/c172.pdf

24

[5] Mohammed Sadraey, “Tail Designs” http://faculty.dwc.edu/sadraey/Chapter%206.%20Tail%20Design.pdf

[6] Lois V. Schmidt, “Introduction to Aircraft Flight Dynamics” (1998 Volume)

[7] Ryan C. Struett, “Empennage Sizing and Aircraft stability using MATLAB”

http://digitalcommons.calpoly.edu/cgi/viewcontent.cgi?article=1074&context=aerosp

[8]Anwar U1 Haque et al., “Estimation of Pitching Moment of a Hybrid Fuselage”

http://www.arpnjournals.org/jeas/research_papers/rp_2016/jeas_0216_3629.pdf

[9] “Dynamic Stability” https://courses.cit.cornell.edu/mae5070/DynamicStability.pdf

[10] Lutze, “Aircraft Dynamics”, http://www.dept.aoe.vt.edu/~lutze/AOE3134/LongitudinalDynamics.pdf

[11] “Red Bull Air Race”, http://www.redbullairrace.com/en_US/article/planes

[12] “Speed of Sound at Different Altitudes”, http://www.fighter-planes.com/jetmach1.htm

[13] FAI, “Edge 540 T 3-View”, http://www.f3m.com/spain/3view/3view.php.html

[14] Robert C. Nelson, “Flight Stability and Automatic Control” 2nd Edition

[15] “Aircraft.cfg ”,file, Microsoft Flight Simulator X.

[16] Jon Berndt, “Selected Aircraft Database”, http://jsbsim.sourceforge.net/aircraft.pdf