Influence of Piston Propeller-driven engine Model into the...

American Institute of Aeronautics and Astronautics

1

Influence of a Piston Propeller-driven Engine Model on the

Design of a Cruise Autopilot

Pedro J. González1

Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, 12228-900, Brazil

Pedro J. Boschetti2

Universidad Simón Bolívar, Naiguatá, Vargas, 1160, Venezuela

and

Flávio J. Silvestre3

Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, 12228-900, Brazil

Piston propeller-driven engines are usually modeled assuming constant brake power (simple

model) to obtain power available. A more realistic model can be used estimating the power

available using a complex procedure. A complex model is proposed to obtain the power

available. Cruise flight autopilots require accurate representations of the aircraft. The aim

of this paper is to evaluate the influence of a piston propeller-driven engine model on the

design of a cruise autopilot for a medium range unmanned aerial vehicle. A cruise flight

autopilot was designed with the capacity to track flight path angle and velocity, and both

propulsion models are used for comparison. A series of simulations was performed to track

each reference separately and together. The simple and complex propulsions models for the

autopilots were capable of tracking the reference signals with a low error; however, for the

simulations, the throttle of the simple model was lower than that for the complex model. It

can be observed that the simple model is generating more thrust available for lower throttle

positions, which reduce the accuracy of the model.

Nomenclature

AJn = coefficients of the advance parameter polynomial curve

APn = coefficients of power available polynomial curve

Aηn = coefficients of the propeller efficiency polynomial curve

CL = lift coefficient

CD = drag coefficient

CP = power coefficient

d = propeller diameter

H = height

J = advance parameter

k = induced drag factor

n = propeller revolutions per second

Pav = power available

Pbrake = brake power

T = thrust

t = time

V = velocity

W = weight of the aircraft

1 PhD Candidate, Aerospace Engineering Division, São José dos Campos, Member AIAA. 2 Associate Professor, Department of Industrial Technology, Camurí Grande Valley, Senior Member AIAA. 3 Assistant Professor, Aerospace Engineering Division, São José dos Campos, Member AIAA.

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AIAA Atmospheric Flight Mechanics Conference

13-17 June 2016, Washington, D.C.

AIAA 2016-3538

Copyright © 2016 by Pedro J. González, Pedro J. Boschetti, Flávio J. Silvestre. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

AIAA Aviation

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2

α = angle of attack

γ = flight-path angle

θ = pitch angle

η = propeller efficiency

ρ = air density at flight altitude

ρo = air density at sea level

σ = density ratio

φ = power-altitude factor

ω = angular velocity of the motor shaft

I. Introduction

HE design of autopilot systems require an accurate mathematical model of the airplane. This model consists of

the aerodynamic model, the inertial-mass model, and the propulsion model. Accuracy of the plant leads to a

better behavior in closed loop.

Textbooks as Stevens and Lewis1 and Zipfel2 present also modelling techniques for the propulsion of the

airplane, and even missiles, must be.2 Zipfel2 discuss on aerodynamic modelling of airplanes and missiles for

rockets, turbojets and turbofans, but not for piston propeller-driven engines, but does not present models for piston

propeller-driven engines.

Usually, the piston propeller-driven engines are modeled assuming constant power and constant propeller

efficiency, and an altitude correction for piston engines is applied.3 Cavcar4 assumes a constant brake power of the

engine (for piston engine or turboprop) and used a polynomial function to estimate the propeller efficiency.

Smetana5 proposes to model the brake power output by a polynomial function and assumed the propeller efficiency

as a constant value. Mises6 explains the process to determine the power available for a piston propeller-driven

engine and suggests a specific equation to estimate the power-altitude factor to consider the variations of output

brake power with altitude. Then, Smetana5 recommends a modified formula for the same aim. Boschetti et al7 model

a piston propeller-driven engine using an algorithm to compute the power available. This model simulates the brake

power by a polynomial function, which is corrected at each time step based on the Mises model for the power-

altitude factor, and the propeller efficiency is calculated using two polynomial functions. To simplify, here the

model described by Anderson3 is called the simple model and the one created by Boschetti et al.,7 the complex

model. The complex model is more difficult to create than the simple model. The first one needs many experimental

data from the engine and the propeller, and it requires a complex algorithm to represent it. The simple model is

represented by a single equation, needing only the maximum brake power at sea level, and the propeller efficiency is

constant.

Questions arise on how the engine model would influence the design of an autopilot system. The main objective

of this paper is to evaluate the influence of a piston propeller-driven engine model into the design of cruise autopilot

for a medium range unmanned aerial vehicle.

II. Propulsion models

The piston propeller driven engine simple model is described by Cavcar4 and Anderson3 as,

VPT av (1)

brakeav PP (2)

where the brake power (Pbrake) and the propeller efficiency (η) are constant, for the same height the power available

is constant (Pav), and the thrust (T) is function of flight velocity (V).

The complex model of a piston propeller-driven engine based on Ref. [6] and presented by Boschetti et al7 uses

as input data the engine charts provided by the manufacturer, propeller charts obtained experimentally, the propeller

diameter, and an atmospheric model for density according to the air temperature and altitude.

The procedure to calculate the available power presented in Ref. [7] it is explained herein. Based on the engine-

propeller configuration, the brake power and the propeller revolutions per second of the engine are arranged in two

vectors, respectively, and the power coefficient (CP) vector for the specific altitude can be computed by,

T

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3

Nidn

PC

i

ibrake

ip,2,1;

53

(3)

where φ is power altitude factor, ρ is the air density at flight altitude, d is the propeller diameter, and n are propeller

revolutions per second.

The power-altitude factor is computed at each time step considering that it is a function of height through the air

density ratio (σ= ρ/ρsea level),8

85.015.0 H (4)

Using the information of propeller charts, polynomial equations are used to express the advance parameter (J) as

a function of power coefficient (Cp), and the propeller efficiency as a function of advance parameter, respectively,

01

2

2

3

3 JPJPJPJPACACACACJ (5)

01

2

2

3

3 AJAJAJAJ (6)

The advanced parameter and the propeller efficiency are expressed in vector form:

PCfJ 1 (7)

Jf2 (8)

The power available and the corresponding velocity is obtained as a function of brake power, efficiency, and

power altitude factor φ as:

NiPP iibrakeiav ,2,1; (9)

NidnJV iii ,2,1; (10)

According to Smetana,5 the power available for a specific altitude may be expressed as a polynomial equation,

01

2

2

3

3 PPPPavAVAVAVAP (11)

Using the method of least squares, the coefficients of the polynomial are computed, based in the values of power

available and flight velocity obtained using Eqs. (9) and (10).

III. Rigid body equations of motion

In this section, the equations of motion of a rigid body aircraft are shown. The flight dynamics of a generic rigid

aircraft is derived and the kinematic equations are presented. The body reference system is used; the equation of

forces and moment, and finally the body angular velocities in terms of the Euler angles and Euler rates are given.

A. Definition of forces and moments for a rigid aircraft The equations of motion are obtained from Newton’s second law; the summation of the external forces acting on

the aircraft is equal to the time rate of change of the momentum of the body. The external moments acting on the

aircraft are equal to the time rate of change of the angular momentum; Eqs. (12)-(13) express these statements,9

mVmVdt

dF b )( (12)

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4

bbb HHdt

dM )( (13)

where F is the vector of forces, m is the mass of the airplane, ω is the angular velocity vector of the body system

respect to the earth but expressed in the body system and Hb=Iω, I is the mass moments of inertia of the aircraft on

each axis.

Once these expressions are developed, the scalar equations are,

qupvwmF

pwruvmF

rvqwumF

z

y

x

(14)

assuming x to the frontal direction of the aircraft, y to the right wing, and z to the ground. The products of inertia

acting on the planes ¨yz¨ and ¨xy¨ are zero, the angular accelerations can be written as

zzxxxz

zzyyxxxzxzxxyyxxxxxz

yy

xzxxzz

zzxxxz

zzyyzzxzzzyyxxxzxzzz

III

qrIIIIpqIIIINILI

I

prIprIIM

III

qrIIIIpqIIIINILI

r

q

p

2

22

22

2

22

)()(

)()(

)()(

(15)

L, M and N are the roll, pitch and yaw moments, respectively. About to the position of the aircraft, the

relationship between the angular velocities in the body frame (p,q,r) and the Euler rates is determined by the next set

of equations;

coscossin0

sincoscos0

sin01

r

q

p

(16)

where ϕ is the bank angle, θ is the pitch angle, and ψ is the yaw angle. This equation could also be expressed in

function of the body angular velocities to calculate the Euler rates,

r

q

p

seccossecsin0

sincos0

tancostansin1

(17)

B. Contribution of forces and moments on the aircraft The equations presented previously can be linearized using the small disturbance theory. The motion of the

aircraft is based on small deviations from the steady flight condition.9

The forces and moments acting on the complete aircraft are defined in dimensionless aerodynamics coefficients.

The aerodynamic coefficients are separated in longitudinal coefficients acting on the lift and drag axes and the

pitching moment primarily dependent on α and on lateral-directional coefficients acting on the sideslip force and on

the β dependent roll and yaw moments.10 Equation (18) defines the angles of incidences and Eq. (19) the true/real

velocity and the dynamic pressure. Equation (20) shows the temporal derivatives of these angles and the true/real

speed. vT represents the lateral velocity and VT the true velocity

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T

T

T

T

V

v

u

warcsin,arctan (18)

2222 )2/1(, TTTTT VqwvuV (19)

T

TTTTTTT

TTT

TTTT

TT

TTTT

V

wwvvuuV

wuV

VvVv

wu

uwwu

,,

2222 (20)

The atmosphere density and the temperature are dependent on the geometric height; the international standard

atmosphere ISA states that:

RTTaHpTaHTT /)/1(),/1( 2561,50000 (21)

where T0=288.15 K, a=–6.5x10–3km–1, R=287.3 m2k–1s–2 and p0=1013×102Nm–2

Adding aerodynamic, thrust, and gravitational forces and moments acting on the aircraft, it is possible to obtain

Eq. (22) and Eq. (23).

coscos

cossin

sin

sin

0

cos

g

g

g

m

F

F

Cl

C

C

LF

F

F

Yw

D

bw (22)

0

0

\ rIFz

bC

cC

bC

SqM FFF

n

M

l

(23)

The thrust force is not only in function of the angle αF between the line of traction and the “Xb” axis. The thrust

of a piston fixed propelled aircraft is a function of throttle δt. Traditionally, it is modeled with Eqs. (1)-(2), and Eq.

(24) includes the throttle percentage.

tV

PT av (24)

In this paper, it is proposed to include the estimation of the power available with Eq. (11). The magnitude of the

aerodynamic coefficients depends on the aerodynamic properties of the vehicle; these coefficients could be

estimated analytically, with numerical methods and simulations, wind tunnel testing or flight tests.1,10

C. Steady flight and trim Steady flight is a flight condition where the total of the forces and moments acting on the aircraft are equal to

zero.10 In order to hold a flight condition, the aircraft requires modifying the position of the control surfaces and

adjusting the throttle. The aerodynamic forces and moment coefficients are a function of the deflection of the control

surfaces: elevator δe, aileron δa, and rudder δr.1,9,10

The state-space modeling techniques were applied in order to simulate the aircraft in flight; this is applicable just

around linearized and steady flight condition. The flight equations may be written around an equilibrium point.

Equation (25) represents the trim flight condition,

0

0)cos(sincossin

0)sin(cossincos

eMe

eeFeeLeeDee

eeFeeLeeDee

ScCq

mgFCCSq

mgFCCSq

(25)

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6

where γe is the flight-path angle in the equilibrium state

eee (26)

The aerodynamic coefficients are a function of δe. They could be estimated for a flight condition. The angle of

trim αe, the deflection of the elevator to hold it, and the amount of throttle to sustain the speed are the results of the

trim calculation.

To represent in state-space variable equations, the selected states and observable variables are presented in Eqs.

(27)-(28).

TDEN xxxrqpwvuX (27)

TzyxT haaarqpVY (28)

The control vector is,

TrcacectcU (29)

The final form of the invariant linear system is:

UBXAX (30)

The linear equation for the outputs of the system is,

UDXCY (31)

Based on this set of equation, a series of scripts were written in Matlab,11 where the dynamics of the rigid motion

of the aircraft is calculated and the states needed for the computation of this dynamics are observable. Once the

model of an aircraft is defined, the steady flight condition may be determined around a fixed speed; the final output

of the trim calculation offers the initial conditions to run the simulations; from this start point, the flight dynamics

evaluation of the aircraft may be completed and the controller designed.

Longitudinal Automatic Pilot

The aim is to design a longitudinal flight control system that will be capable of holding the flight speed and the

flight-path. This is done by regulating some states of the aircraft to zero while the desirable close-loop response is

obtained.12 The control system computes multiple inputs and multiple outputs and then it derives a matrix equation

that solves the control gains.1,12

The control technique used was LQR (linear quadratic regulator), which consists in minimizing a quadratic cost

or performance index J.

02

1dtRuuQxxJ TT (32)

For a system of the form,

Kxu

Cxy

BuAxx

(33)

The result of the closed loop system is

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7

xAxBKCAx c )( (34)

The output feedback gain K necessary to minimize J is obtained by the algebraic Ricatti equation

PBPBRQPAPA TT 10 (35)

The weighting matrix Q=HT’*HT. These HT matrices define the tracked output. R is an identity matrix of the size

of the control inputs. Then, the Kalman gain is calculated as

PBRK T1 (36)

Assuming that the close loop system is stable, the performance index becomes

)0()0(2

1PxxJ T (37)

Figure 1 shows the structure of the control system proposed for this work. The outer loop is based on classical

flight control outer loops for flight path angle and velocity. The elevator controls the flight path angle and the engine

the changes in velocity.

Case Study

To evaluate different engine models in the cruise flight autopilot, the Unmanned Aircraft for Ecological

Conservation (ANCE) was selected. The ANCE is a reconnaissance-unmanned aircraft for the detection of oil

leakages. The aircraft was selected because enough information of the airplane is available. Table 1 shows the

characteristics of the ANCE.7

The modeled engine is the Simonini Victor 1 Plus13 with a propeller 5868-9, Clark Y section, two blades with

blade angle of 20 deg and a diameter of 0.947 m.14 Table 2 shows the characteristic of the engine, and Fig. 2

presents the propeller charts.

Table 3 presents the aerodynamic model of the ANCE. The flight condition was linearized and the steady state

flight arrangement of the aircraft was determined for a speed Vto=54 m/s and a height of 2438 m. The simulations

will start from the steady state condition presented in table 4.

The automatic control system for an aircraft must have these functionalities: a feedback output to increase the

stability of the internal system, which could be identified as a stability augmentation system, and a dynamic

controller for tracking outer loop variables; this is carried out with a proportional integral lead-lag compensator for

the flight path angle and a lead-lag for the velocity. The structure of the compensators is dependent on the dynamics

of the actuator; they are independent parameters of each project. The transfer function for the close loop system for

the flight-path angle and the true velocity compensator are presented as Eq. (38). From a Bode diagram, the phase

margins of the closed loop system were -27.4 dB and 60.8 dB for the flight path angle tracker; the close loop is

stable.

Aircraft

Actuator

C-K

-H

-LCompensator

x

xy

u

-z

r e v

Figure 1. Structure of the control system

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8

5

7697.63.1,

10.

339.11.25506.22

2

s

sTf

ss

ssTf Vg (38)

Table 1. Characteristics of the ANCE

Dimensions

Longitude 4.648 m

Height 1.116 m

Span 5.187 m

Chord 0.604 m

Surface 3.1329 m2

Weight 182.055 kg

Inertia Matrix 2mkg

40000

04000

00150

Performance

Cruise speed 54 m/s

Take-off distance 270 m

Engine Simonini Victor 1 Plus

Propeller 5868-9, Clark Y section, two

blades with blade angle of 20 deg

Table 2. Characteristics of the Simonini Victor 1

Plus engine.13

ω, rev/min Pbrake, hp

3000 15.6

3200 17.28

3500 20.3

4000 25.6

4500 31.5

5000 36.65

5500 41.8

6000 45.6

6200 47.12

6500 47.45

6700 45.6

Figure 2. Curves of propeller 5868-9, Clark Y

section, two blades with blade angle of 20 deg.7,14

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9

Having these specifications, the linear model of the aircraft is defined and the model could be built in Simulink

to make close-loop simulations. The exogenous references are presented as tracking commands, to hold speed and

flight-path.

The gain vector was tuned with Matlab.11 The routine fmincon was used to obtain the optimal gain. This routine

finds a constrained minimum of a scalar function of several variables starting at an initial estimate. This is generally

referred to as constrained nonlinear optimization or nonlinear programming minimum of constrained nonlinear

multivariable function.

The control system was estimated with the complex model. Then, the simulation was performed with both

models and compared. Figure 3 and 4 are the result of a 30 s simulation tracking an increase of velocity from 54 m/s

to 58 m/s. The measured outputs (Fig. 3) are the velocity, the flight path angle, the height, pitch rate, angle of attack,

and pitch angle. Figure 4 shows the variation of elevator and throttle commands. It is possible to observe the

difference between the response of both controller to attain commanded speed. In fact, during the 30 s of simulation

the simple model controller does not attain the commanded and has a final error of 32 percent. The commanded

flight path angle is satisfactorily kept in both cases. Pitch rate and height seems to be changing similarly but at

different rates. Angle of attack and pitch angle are different during the entire simulation.

When engine models are compared, it is observed how the complex model holds full throttle for more than 4 s

while the aircraft is accelerating. There is a difference of 7percent in the final throttle obtained for both engines; the

simple model appears to be producing more thrust at lower throttle to hold the commanded speed. The saturation

limits applied to the complex model are not compatible with the simple model. It is producing more thrust in the

initial section of the maneuver.

A second evaluation was performed to track a flight path angle of two degrees. Figure 5 shows the variation of

the flight path angle until it attains the command; the increase of height associated with this maneuver is also

appreciable. There is a difference of 0.07 m/s between the velocities of both models, which represent a difference

less that 1 percent on stationary state error. Meanwhile, the rest of the monitored variables behave similarly in both

cases. Figure 6 presents the commanded inputs during the maneuver while the elevator seems to have a very similar

deflection for both cases. The throttle shows a difference of 12 percent between the two models.

Table 3. Aerodynamic model of the ANCE

Forces

)2/(22089.642018.53273.0 0TL VqcC

20488.00266.0 LD CC

)2/(421165.0)2/(00483.0002865.02807.032659.0 00 TTY VrbVpbarC

tVPF Toav )/(

Moments

raTTl VrbVpbC 042972017419.0)2/(210582.0)2/(12683.113751.0 00

)2/(6627.18687172.283882.30959.0 0TeM VqcC

raTTn VrbVpbC 017189.00292208.0)2/(52642.0)2/(0322.0189076.0 00

Table 4. Steady flight condition for the aircraft

Velocity 54 m/s

Height 2438 m

Pitch angle 0.754 deg

Throttle simple model 38.25%

Throttle complex model 47.04%

Elevator deflection -3.118 deg

Aileron deflection 0 deg

Rudder deflection 0 deg

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10 .

0 10 20 3054

56

58

60

Time, s

Vt,

m/s

Simple model

Complex model

0 10 20 302438

2438.5

2439

Time, s

Heig

ht,

m

0 10 20 30-0.1

0

0.1

0.2

0.3

Time, s

, d

eg

0 10 20 30-0.3

-0.2

-0.1

0

0.1

Time, s

q, ra

d/s

0 10 20 300

0.2

0.4

0.6

0.8

Time, s

, deg

0 10 20 300

0.2

0.4

0.6

0.8

Time, s

, deg

Figure 3. Measured Outputs for velocity tracking

0 10 20 30-3.5

-3

-2.5

-2

Time, s

e

, deg

Simple model

Complex model

0 10 20 300

0.5

1

1.5

Time, s

t

Figure 4. Elevator and throttle commands

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The third case consists in an increase of velocity of 2 m/s and a commanded flight path angle of 1 deg. Fig. 7

shows how the ANCE tracks the commanded flight path angle and starts to climb while the velocity increases. Once

again, Fig. 8 shows that there is a discrepancy between the throttle of the simple model, which is providing with

more thrust at lower throttle positions than the complex model. The difference is 10 percent for the stationary

condition.

It appears that the simple model is generating more power for the same throttle position than the complex model.

The tracking commands were achieved using both models, although the error with the simple model was about 5

percent on average. Once again the dynamics of the simple model does not respect the same boundaries applied to

the complex model which cause and unfeasible situation for the first second of the maneuver.

In this flight conditions, the utilization of both models seems not to influence drastically in the result of the

maneuvers, except in the first case the variation from the trim condition was completely responsibility of the engine.

However, there exists consistent inaccuracy between the throttle level required for each maneuver.

0 10 20 3054

54.1

54.2

54.3

Time, s

Vt,

m/s

Simple model

Complex model

0 10 20 302420

2440

2460

2480

2500

Time, s

Heig

ht,

m

0 10 20 300

1

2

3

Time, s

, d

eg

0 10 20 30-1

0

1

2

3

Time, s

q,

rad

/s

0 10 20 30

0.8

1

1.2

1.4

Time, s

,

deg

0 10 20 300

1

2

3

4

Time, s

,

deg

Figure 5. Measured Outputs for flight path angle tracking

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12

0 10 20 3054

55

56

57

Time, s

Vt,

m/s

Simple model

Complex model

0 10 20 302430

2440

2450

2460

2470

Time, s

Heig

ht,

m

0 10 20 300

0.5

1

1.5

Time, s

, d

eg

0 10 20 30-0.5

0

0.5

1

1.5

Time, s

q, ra

d/s

0 10 20 300.4

0.6

0.8

1

Time, s

,

deg

0 10 20 300.5

1

1.5

2

Time, s

, d

eg

Figure 7. Measured outputs for velocity and flight path angle tracking

0 10 20 30-8

-6

-4

-2

0

Time, s

e

, d

eg

Simple model

Complex model

0 10 20 300.4

0.5

0.6

0.7

0.8

Time, s

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Summary

A control system for longitudinal cruise flight was projected to evaluate the influence of two different piston

fixed propeller engine models. Both engine models were described as well as the aircraft rigid model and the control

structure. Three cases of simulation were carried out using the ANCE. The complex model is capable of providing a

more accurate information of the power available for the engine and as result of the provided thrust. The simple

model seems to develop more thrust at lower throttle position than the complex model in all simulation cases, which

could be an inaccuracy of the model. In flight conditions that precise more power available the discrepancy between

engine models really affects the performance of the autopilot and commands an unfeasible situation. The autopilot

seems to be capable of tracking all commanded references with both models with small errors. The simple model

does not represent accurately enough the performance of the engine what affect directly on the expected

performance of the aircraft.

References 1Stevens, B. L., and Lewis, F. L. Aircraft Control and Simulation, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. 2Zipfel, P. H. “Three-Degrees-of-Freedom Simulation,” Modeling and Simulation of Aerospace Vehicle Dynamics, 2nd ed.,

AIAA Education Series, AIAA, Reston, VA, 2007. 3Anderson, J. D., Aircraft Performance and Design, WCB/McGraw-Hill, Boston, 1999, Chaps. 5. 4Cavcar, A., “Climb Performance of Piston–Propeller Airplane with Cambered Wing and Variable Propeller Efficiency,”

Journal of Aircraft, Vol. 48, No. 5, 2011, pp. 1701-1707. 5Smetana, F. O., “Flight Vehicle Performance,” Flight Vehicle Performance and Aerodynamic Control, AIAA Education

Series, AIAA, Reston, VA, 2001, pp. 145-152, 162-165, 171-181, 185-191. 6Mises (von), R., “The general performance problem,” Theory of Flight, Dover Publications, New York, 1959, pp. 385-394. 7Boschetti, P. J., González, P. J., and Cárdenas, E. M. “Program to Calculate the Performance of Airplanes Driven by a

Fixed-Pitch Propeller,” AIAA paper 2015-0015, 2015. 8Mises (von), R., “The airplane engine,” Theory of Flight, Dover Publications, New York, 1959, pp. 354,365. 9Nelson, R. C., “Longitudinal Motion,” Flight Stability and Automatic Control, 2nd ed., McGraw Hill, Boston, MA, 1998 10Etkin, B., Dynamics of Atmospheric Flight, John Wiley & Sons, Inc., New York City, NY, 1972. 11MATLAB, The MathWorks, Inc., Software Package, Ver. 7.8.0 (R2013a), 2013. 12McLean, D, Automatic Flight Control Systems, Prentice Hall, Inc, Southampton, Hampshire, UK, 1990. 13Simonini Flying, VICTOR 1 PLUS [online database], URL: http://www.simonini-flying.com/victor1plus_eng.htm [cited 13

September 2013]. 14Hartman, E. P., and Biermann, D., “The aerodynamic characteristics of full-scale propellers having 2, 3, and 4 blades of

Clark Y and R.A.F. 6 airfoil sections,” NACA TR-640, 1938.

0 10 20 30-5

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http://www.simonini-flying.com/victor1plus_eng.htm

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