Linearized Longitudinal Equations of Motion

[Pages:29]11/13/18

Linearized Longitudinal Equations of Motion

Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018

Learning Objectives

? 6th-order -> 4th-order -> hybrid equations ? Dynamic stability derivatives ? Long-period (phugoid) mode ? Short-period mode

Reading: Flight Dynamics 452-464, 482-486

Copyright 2018 by Robert Stengel. All rights reserved. For educational use only.



1

6th-Order Longitudinal

Fairchild-Republic A-10

Nonlinear Dynamic Equations

Equations of Motion

Symmetric aircraft Motions in the vertical plane

Flat earth State Vector, 6 components

u! = X / m - gsin - qw

w! = Z / m + g cos + qu

x!I = (cos )u + (sin )w z!I = (- sin )u + (cos )w

q! = M / Iyy

! = q

u w x z q

=

Axial Velocity Vertical Velocity

Range Altitude(?) Pitch Rate Pitch Angle

x1

x2

x3 x4 x5

=

x Lon6

x6

Range has no dynamic effect Altitude effect is minimal (air density variation)

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4th-Order Longitudinal Equations of Motion

Nonlinear Dynamic Equations, neglecting range and altitude

u! = f1 = X / m - gsin - qw w! = f2 = Z / m + g cos + qu q! = f3 = M / Iyy

! = f4 = q

State Vector, 4 components

x1

x2 x3

=

x Lon4

x4

u w q

=

Axial Velocity, m/s Vertical Velocity, m/s

Pitch Rate, rad/s Pitch Angle, rad

3

Fourth-Order Hybrid Equations of Motion

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2

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Transform Longitudinal Velocity Components

Replace Cartesian body components of velocity by polar inertial components

u! = f1 = X / m - gsin - qw w! = f2 = Z / m + g cos + qu q! = f3 = M / Iyy ! = f4 = q

x1 x2 x3 x4

=

u w q

=

Axial Velocity Vertical Velocity

Pitch Rate Pitch Angle

5

Transform Longitudinal Velocity Components

Replace Cartesian body components of velocity by polar inertial components

V! = f1 = T cos( + i) - D - mgsin m ! = f2 = T sin( + i) + L - mg cos mV

q! = f3 = M / Iyy

! = f4 = q

Hawker P1127 Kestral

x1 x2 x3 x4

=

V

q

=

Velocity Flight Path Angle

Pitch Rate Pitch Angle

i = Incidence angle of the thrust vector with respect to the centerline

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3

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Hybrid Longitudinal Equations of Motion

? Replace pitch angle by angle of attack = -

V! = f1 = T cos( + i) - D - mgsin m ! = f2 = T sin( + i) + L - mg cos mV

q! = f3 = M / Iyy

! = f4 = q

x1 x2 x3 x4

=

V

q

=

Velocity Flight Path Angle

Pitch Rate Pitch Angle

7

Hybrid Longitudinal Equations of Motion

? Replace pitch angle by angle of attack = -

V! = f1 = T cos( + i) - D - mgsin m ! = f2 = T sin( + i) + L - mg cos mV

q! = f3 = M / Iyy

!

= ! - !

=

f4

=

q-

f2

=

q-

1 mV

T

sin (

+ i) +

L-

mg cos

x1 x2 x3 x4

=

V

q

=

Velocity Flight Path Angle

Pitch Rate Angle of Attack

= +

8

4

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Why Transform Equations and State Vector?

x1 x2 x3 x4

=

V

q

=

Velocity Flight Path Angle

Pitch Rate Angle of Attack

Velocity and flight path angle typically have slow variations

Pitch rate and angle of attack typically have quicker variations

Coupling typically small

9

Small Perturbations from Steady Path

x!(t) = x! N (t) + x!(t) f[xN (t), uN (t), wN (t),t]

+ F(t )x(t) + G(t )u(t) + L(t )w(t)

Steady, Level Flight

x! N (t) 0 f[xN (t), uN (t), wN (t),t] x!(t) Fx(t) + Gu(t) + Lw(t)

Rates of change are "small"

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Nominal Equations of Motion in Equilibrium (Trimmed Condition)

x N (t) = 0 = f[xN (t), uN (t), wN (t),t]

T

xTN

=

# $

VN

N

0

N

% &

= constant

T, D, L, and M contain state, control, and disturbance effects

V!N = 0 = f1 = T cos( N + i) - D - mgsin N m !N = 0 = f2 = T sin( N + i) + L - mg cos N mVN

q!N = 0 = f3 = M Iyy

! N

=

0

=

f4

= (0)-

1 mVN

T

sin( N

+ i) +

L - mg cos

N

11

Flight Conditions for

Steady, Level Flight

Nonlinear longitudinal model

V

=

f1

=

1 m $%T

cos(

+ i)-

D-

mg sin

&'

=

f2

=

1 mV

$%T sin(

+ i)+

L - mg cos

&'

q = f3 = M / Iyy

=

f4

= -

=

q-

1 mV

$%T sin(

+ i)+

L

- mg cos

&'

Nonlinear longitudinal model in equilibrium

0

=

f1

=

1 m

$%T

cos(

+ i)-

D - mg sin

&'

0

=

f2

=

1 mV

$%T sin(

+ i)+

L - mg cos

&'

0 = f3 = M / Iyy

0

=

f4

= -

=

q-

1 mV

$%T

sin (

+ i)+

L-

mg cos

&'

12

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Numerical Solution for Level Flight Trimmed Condition

? Specify desired altitude and airspeed, hN and VN ? Guess starting values for the trim parameters, T0, E0, and 0

? Calculate starting values of f1, f2, and f3

f1

=

V!

=

1 m

T

( T

,

E,,

h,V

)cos(

+

i)

-

D ( T

,

E,,

h,V

)

f2

=

!

=

1 mVN

T

( T

,

E, , h,V

)sin(

+

i)+

L(T ,

E, , h,V

)-

mg

f3 = q! = M (T , E,,h,V ) / Iyy

? f1, f2, and f3 = 0 in equilibrium, but not for arbitrary T0, E0, and 0 ? Define a scalar, positive-definite trim error cost function, e.g.,

J (T ,E, ) = a( f12 ) + b( f22 ) + c( f32 )

13

Minimize the Cost Function with Respect to the Trim Parameters

Error cost is bowl-shaped

J (T ,E, ) = a( f12 ) + b( f22 ) + c( f32 )

Cost is minimized at bottom of bowl, i.e., when

$ &

J

J

J

' )=0

% T E (

Search to find the minimum value of J

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Example of Search for Trimmed Condition (Fig. 3.6-9, Flight Dynamics)

In MATLAB, use fminsearch or fsolve to find trim settings

(T *,E*, *) = fminsearch#$J,(T ,E, )%&

15

Small Perturbations from Steady Path Approximated by Linear Equations

Linearized Equations of Motion

x! Lon

=

V! !

q! !

=

FLon

V

q

+

G

Lon

T E

"

+

"

16

8

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