an introduction to flight dynamics

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

University of São PauloSão Carlos School of Engineering

Department of Aeronautical Engineering

An introduction to flight dynamicsFlight Dynamics II

Ricardo Afonso Angélicoricardoangelico@gmail.com

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Human factors

Symptoms 2 < f < 20 Hz● Acceleration● Vibration● Combined stresses

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

University of São PauloSão Carlos School of Engineering

Department of Aeronautical Engineering

General equations unsteady motionFlight Dynamics II

Ricardo Afonso Angélicoricardoangelico@gmail.com

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Model

dm

z

zE

x

y

yE

xE

OE

OB

v

u

w

v

wVBrCE

rE

X

Y

Z

N

M

L

p

q

r

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Aircraft orientation - Euler angles

0

1

2

3

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Motion equations

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Motion equations

Equations in expanded form:

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Trimmed flight

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Small disturbance theory

Procedure:

1) Substituir nas equações de movimento2) Desprezar o produto de pequenas perturbações3) Subtrair as equações de voo trimado4) Utilizar aproximações de senos e cossenos para pequenos ângulos

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Small disturbance theory

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Voo retilineo

Small disturbance theory

LongitudinalLateral

Motion decomposition: longitudinal / lateral

● Existence of a plane of symmetry● The absence of rotor gyroscopic effects

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Linear motion equations

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Longitudinal – Small disturbance equations

Elevator

Propulsion

Etkins notation

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Lateral – Small disturbance equations

Etkins notation

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Euler's formula

Control fixed problems

Considering a conjugatepair of eigenvalues

Oscillatory mode of period

Decays depends on

Pair of terms eigenvector

eigenvalue

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Eigenvalues interpretationEigenvalues

real complex

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Eigenvalues interpretationEigenvalues

real complex

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Eigenvalues interpretation

Important parameters

● Period● Time to double of time to half● Cycles to double or cycles to half

Until now, we cannot inferabout aircraft handling qualities

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Eigenvalues interpretation

Time to double or half

Cycles to double or half

Logarithmic decrement (log ratio of successive peaks)

“undamped” circular frequency

damping ratio

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Longitudinal modes of an aircraftBoeing 747 at Mach number 0.8 (40000 ft) cruising in horizontal flight

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Longitudinal modes of an aircraft

Computing eigenvalues in MATLAB[a,b] = eig(A)

Boeing 747 – Longitudinal matrix

Mode Name Period(s)

thalf(s)

Nhalf (cycles)

1 Phugoid 93.4 211 22.52 Short-period 7.08 1.86 0.26

Phugoid mode was first described by Lanchester in 1908

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Longitudinal modes of an aircraftBoeing 747 – Longitudinal matrix

Mode Name Period(s)

thalf(s)

Nhalf (cycles)

1 Phugoid 93.4 211 22.52 Short-period 7.08 1.86 0.26

Phugoid mode was first described by Lanchester in 1908

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Longitudinal modes of an aircraftBoeing 747 – Phugoid motion

Mode Name Period(s)

thalf(s)

Nhalf (cycles)

1 Phugoid 93.4 211 22.52 Short-period 7.08 1.86 0.26

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Longitudinal modes of an aircraftBoeing 747 – Short period

Mode Name Period(s)

thalf(s)

Nhalf (cycles)

1 Phugoid 93.4 211 22.52 Short-period 7.08 1.86 0.26

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Reduced order models

Short mode approximation

Rewriting...

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Reduced order models

For Boeing 747:

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Reduced order modelsPhugoid mode approximation

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Reduced order modelsPhugoid mode approximation

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Reduced order modelsPhugoid mode approximation

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Reduced order modelsPhugoid mode approximation

Solving the equation:

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Exercises

Roskam notation

Tasks

● Compute● the eigenvalues● Period● Time to double of time to half● Cycles to double or cycles to half

● Repeat the tasks using a reduced model for short period

Airplane A Airplane B

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Exercises

Airplane A – Cessna 182 Airplane B – Cessna T-37 A

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

University of São PauloSão Carlos School of Engineering

Department of Aeronautical Engineering

Lateral motionFlight Dynamics II

Ricardo Afonso Angélicoricardoangelico@gmail.com

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Control fixed problems

No control forces

First-order differential equation

Solution

Eigenvalues are given by:

Solution:

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Lateral motion eigenvalues

Eigenvalues expected:● Two reals eigenvalues corresponding to spiral mode and rolling

convergence● One complex conjugate pair corresponding to dutch-roll mode

For real eigenvalues, a characteristic time can be computed:: real eigenvalues

: characteristic time for spiral mode

: characteristic time for rolling convergence

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Lateral motion equations

Main lateral modes:● Spiral mode● Rolling mode● Dutch roll

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

University of São PauloSão Carlos School of Engineering

Department of Aeronautical Engineering

Transfer functionsFlight Dynamics II

Ricardo Afonso Angélicoricardoangelico@gmail.com

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Transfer functions

State-space form:

are matrices of constant coefficients

is the Transfer Function Matrix

is the polynomial matrix of function numerators

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Transfer functions

Considering

Numerator matrix

Characteristic polynomial

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

University of São PauloSão Carlos School of Engineering

Department of Aeronautical Engineering

Pilot ratings andaircraft handling qualities

Flight Dynamics II

Ricardo Afonso Angélicoricardoangelico@gmail.com

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Introduction

✔ Provide some description of requirements on the aircraft dynamics

✔ VLA – Very light airplanes (JAR-VLA)

✔ FAR-23 (Wto < 5000 lb)

✔ FAR-25 (Wto > 5000 lb)

✔ MIL-F-8785C – military aircraft

FlightDynamics

System control andControl actuation systems

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Flight envelope

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Flight envelope

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Flight envelope

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Flight envelope

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Levels of aircraft qualities● Aircraft control authority

● To maintain steady-state rectilinear flight (ex.: cruise conditions)

● To maintain steady-state maneuvering(ex.: circular holding pattern)

● To take-off and to climb safely● To conduct an approach and to land safely

● Pilot compensation● Mental effort by the pilot while at the controls of aircraft

● Pilot workload● Maximum required stick force● Stick force per g● Stick force-speed gradient● Maximum aileron wheel force● Reversible and irreversible control systems

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Reversible control system

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Reversible control system

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Irreversible control system

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Levels of flying qualities● Classification

● Sublevels

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Cooper-Harper Pilot Rating Scale (CHPRS)

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Classes of aircraft

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Classes of aircraft

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Classification of aircraft maneuvers

● Category A

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Classification of aircraft maneuvers● Category B

● Category C

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Flying qualities for longitudinal dynamics● Longitudinal control forces

● Control forces for steady-state flight condition● Control forces for maneuvered flight● Control forces for take-off and landing● Control forces during dives

● Control forces for steady-state flight condition

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Flying qualities for longitudinal dynamics● Control forces for maneuvered flight

● Control force vs. load factor gradient

● Civil aircraft

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Flying qualities for longitudinal dynamics● Control forces for maneuvered flight

● Military aircraft

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Flying qualities for longitudinal dynamics● Control forces for take-off and landing

● Control forces for dives

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

'Requirements'● Phugoid mode

● Short-period mode

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Flying qualities for lateral directional● Lateral directional control forces

● Control forces on rolling maneuvers● Control forces for holding an specific direction (heading) with

asymmetric loading configurations● Control forces for holding an specific direction (heading) with

bank angle with an engine-out condition

● Control forces on rolling maneuvers

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Flying qualities for lateral directional● Control forces on rolling maneuvers

● Control forces for holding heading with asymmetric loading

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

Flying qualities for lateral directional● Control forces on rolling maneuvers

● Control forces for holding heading with asymmetric loading

● Control forces for holding an specific direction (heading) with bank angle with an engine-out condition

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

'Requirements'● Dutch-roll mode

University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A.

References

✔ Etkin, B.; Reid, L. D. (1996) Dynamics of Flight: Stability and Control. John Wiley & Sons. 3rd ed.

✔ Roskam, J. (2001) Airplane Flight Dynamics and Automatic Flight Controls. DARcorporation.

✔ Davis, J. R.; Johnson, R.; Stepanek, J.; Fogarty, J. A. (2008) Fundamental of Aerospace Medicine. Lippincott Williams & Wilkins. 4th ed.

✔ Napolitano, M. R. (2011) Aircraft Dynamics: from modeling to simulation. John Wiley & Sons. Chapter 10 – Pilot rating and aircraft handling