An introduction to flight dynamics
65
University of São Paulo – São Carlos School of Engineering – Department of Aeronautical Engineering ANGÉLICO, R.A. University of São Paulo São Carlos School of Engineering Department of Aeronautical Engineering An introduction to flight dynamics Flight Dynamics II Ricardo Afonso Angélico [email protected]
Transcript of 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 Angé[email protected]
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 Angé[email protected]
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 Angé[email protected]
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 Angé[email protected]
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 Angé[email protected]
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
