Linearized Equations of Motion!
Robert Stengel, Aircraft Flight Dynamics!
MAE 331, 2016
Learning Objectives
Develop linear equations to describe
small perturbational motions
Apply to aircraft dynamic equations
Reading:!
Flight Dynamics!
234-242, 255-266, 274-297, 321-325, 329-330!
Copyright 2016 by Robert Stengel. All rights reserved. For educational use only.
http://www.princeton.edu/~stengel/MAE331.html
http://www.princeton.edu/~stengel/FlightDynamics.html
1
Assignment #6"
due: End of day, Nov. 22, 2016!
•! Code the aerodynamic, thrust, and inertial
properties of the HondaJet in AeroModel.m
(MATLAB function for FLIGHTver2.m)
2
Review Questions!
!! Describe the functions of airplane control systems:!
!!
!!
!!
!!
Elevator/Stabilator!
Ailerons/Elevons!
Rudder!
Spoilers!
!! How can you control the yaw dynamics of a flying
wing airplane?!
!! What are compensating ailerons?!
!! Why aren’t all control surfaces “all moving”?!
!! How are the mechanical dynamics of control systems
modeled?!
!! What is a horn balance, and how does it work?!
!! What are control tabs, and how do they work?!
3
How Is System
Response Calculated?
•! Linear and nonlinear, time-varying and timeinvariant dynamic models
–! Numerical integration (
time domain
)
•! Linear, time-invariant (LTI) dynamic models
–! Numerical integration (
time domain
)
–! State transition (
time domain
)
–! Transfer functions (
frequency domain
)
4
Integration Algorithms
T
x (T ) = x ( 0 ) + % f !" x ( t ) ,u ( t ) ,w ( t ) #$dt
•! Exact
0
•! Rectangular (Euler) Integration
x(t k ) = x(t k!1 ) + " x(t k!1 ,t k )
# x(t k!1 ) + f [ x(t k!1 ),u(t k!1 ),w(t k!1 )] " t
" t = t k ! t k!1
•! Trapezoidal (modified Euler) Integration (~MATLABs ode23)
1
[# x1 + # x 2 ]
2
where
x(t k ) ! x(t k"1 ) +
# x1 = f [ x(t k"1 ),u(t k"1 ),w(t k"1 )] # t
# x 2 = f [ x(t k"1 ) + # x1 ,u(t k ),w(t k )] # t
See MATLAB manual for descriptions of ode45 and ode15s
5
Numerical Integration:
MATLAB Ordinary Differential Equation
Solvers*
•!
Explicit Runge-Kutta
Algorithm
•!
Adams-Bashforth-Moulton
Algorithm
•!
Numerical Differentiation
Formula
•!
Modified Rosenbrock
Method
•!
Trapezoidal Rule
•!
Trapezoidal Rule w/Back
Differentiation
* http://www.mathworks.com/access/helpdesk/help/techdoc/index.html?/access/helpdesk/help/techdoc/ref/ode23.html.
Shampine, L. F. and M. W. Reichelt, "The MATLAB ODE Suite," SIAM Journal on Scientific Computing, Vol. 18, 1997, pp 61-22.
Nominal and Actual Trajectories
•! Nominal (or reference) trajectory and
control history
{x N (t), u N (t), w N (t)}
for t in [t o ,t f ]
x : dynamic state
u : control input
w : disturbance input
•! Actual trajectory perturbed by
–! Small initial condition variation, !xo(to)
–! Small control variation, !u(t)
{x(t), u(t),w(t)}
for t in [t o ,t f ]
= {x N (t) + !x(t), u N (t) + !u(t),w N (t) + !w(t)}
7
Both Paths Satisfy the
Dynamic Equations
Dynamic models for the actual and the
nominal problems are the same
x! N (t) = f[x N (t),u N (t),w N (t)], x N ( t o ) given
x! (t) = f[x(t),u(t),w(t)], x ( t o ) given
Differences in initial
condition and forcing ...
!x(t o ) = x(t o ) " x N (t o )
#% !u(t) = u(t) " u N (t)
$
%& !w(t) = w(t) " w N (t)
'%
( in *+t o ,t f ,%)
... perturb rate of change
and the state
"$ x! (t) = x! N (t) + !!x(t)
#
$% x(t) = x N (t) + !x(t)
&$
' in )*t o ,t f +,
$(
8
Approximate Neighboring
Trajectory as a Linear Perturbation
to the Nominal Trajectory
x! N (t) = f[x N (t),u N (t),w N (t),t]
x! (t) = x! N (t) + !!x(t) = f[x N (t) + !x(t),u N (t) + !u(t),w N (t) + !w(t),t]
Approximate the new trajectory as the sum of the
nominal path plus a linear perturbation
x! (t) = x! N (t) + !!x(t)
#f
#f
#f
" f[x N (t),u N (t),w N (t),t] +
!x(t) +
!u(t) +
!w(t)
#x
#u
#w
9
Linearized Equation Approximates
Perturbation Dynamics
•! Solve for the nominal and perturbation trajectories
separately
Nominal Equation
x! N (t) = f[x N (t), u N (t), w N (t),t], x N ( t o ) given
Perturbation Equation
dim(x) = n ! 1
dim(u) = m ! 1
dim(w) = s ! 1
$
' $
' $
'
&# f
) &# f
) &#f
)
!w(t))
!!x(t) " & x=x (t ) !x(t)) + & x=x (t ) !u(t)) + &
x=x
(t
)
# x NN (t )
# u NN (t )
# w u=uNN (t )
&% u=u
)( &% u=u
)( &%
)(
w=w N (t )
w=w N (t )
w=w N (t )
dim(!x) = n " 1
dim(!u) = m " 1
dim(!w) = s " 1
" F(t)!x(t)+ G(t)!u(t)+ L(t)!w(t), !x (t o ) given
10
Jacobian Matrices Express Solution
Sensitivity to Small Perturbations
Stability matrix, F, is square
"
$
$
!f
F(t) =
=$
x=x N (t )
! x u=uN (t ) $
w=w N (t )
$
$
dim(F) = n ! n
#
! f1
! f1 %
!
'
! x1
! xn '
! ! ! '
'
! fn
! fn '
!
! x1
! xn ' x=x N (t )
&
u=u N (t )
w=w N (t )
11
Sensitivity to Control Perturbations, G
"
$
$
!f
G(t) =
=$
x=x N (t )
! u u=u
$
N (t )
w=w N (t )
$
$
#
dim(G) = n ! m
! f1
! f1 %
!
'
! u1
! um '
! ! ! '
'
! fn
! fn '
!
! u1
! um ' x=x N (t )
&
u=u N (t )
w=w N (t )
12
Sensitivity to Disturbance
Perturbations, G
"
$
$
!f
L(t) =
=$
x=x N (t )
! w u=u
$
N (t )
w=w N (t )
$
$
#
! f1
! f1 %
!
'
! w1
! ws '
! ! ! '
'
! fn
! fn '
!
! w1
! ws ' x=x N (t )
&
dim(L) = n ! s
u=u N (t )
w=w N (t )
13
Scalar Example of Nominal and
Perturbation Equations
Actual System
x! ( t ) = x ( t ) + 2x 2 ( t ) + 3u ( t ) + 4w 3 ( t )
Nominal System
x! N ( t ) = x N ( t ) + 2x N 2 ( t ) + 3u N ( t ) + 4wN 3 ( t )
Perturbation System
!!x ( t ) = !x ( t ) + 4x N ( t ) !x ( t ) + 3!u ( t ) + 12wN2 ( t ) !w ( t )
14
Comparison of Damped
Linear and Nonlinear Systems
Linear Spring
x!1 (t) = x2 (t)
x!2 (t) = !10x1 (t) ! x2 (t)
Spring
Linear Spring Force vs.
Displacement
Displacement
Rate of Change
Damper
Linear plus Stiffening Cubic Spring
x!1 (t) = x2 (t)
Cubic Spring Force vs.
Displacement
x!2 (t) = !10x1 (t) ! 10x13 (t) ! x2 (t)
Linear plus Weakening Cubic Spring
x!1 (t) = x2 (t)
Cubic Spring Force vs.
Displacement
x!2 (t) = !10x1 (t) + 0.8x13 (t) ! x2 (t)
15
MATLAB Simulation of Linear and
Nonlinear Dynamic Systems
MATLAB Main Script
% Nonlinear and Linear Examples
clear
tspan = [0 10];
xo = [0, 10];
[t1,x1 = ode23('NonLin',tspan,xo);
xo = [0, 1];
[t2,x2] = ode23('NonLin',tspan,xo);
xo = [0, 10];
[t3,x3] = ode23('Lin',tspan,xo);
xo = [0, 1];
[t4,x4] = ode23('Lin',tspan,xo);
subplot(2,1,1)
plot(t1,x1(:,1),'k',t2,x2(:,1),'b',t3,x3(:,1),'r',t4,x4(:,1),'g')
ylabel('Position'), grid
subplot(2,1,2)
plot(t1,x1(:,2),'k',t2,x2(:,2),'b',t3,x3(:,2),'r',t4,x4(:,2),'g')
xlabel('Time'), ylabel('Rate'), grid
Linear System
x˙1 (t) = x 2 (t)
x˙ 2 (t) = "10x1 (t) " x 2 (t)
function xdot = Lin(t,x)
% Linear Ordinary Differential Equation
% x(1) = Position
% x(2) = Rate
xdot = [x(2)
-10*x(1) - x(2)];
Nonlinear System
x˙1 (t) = x 2 (t)
3
x˙ 2 (t) = "10x1 (t) + 0.8x1 (t) " x 2 (t)
function xdot = NonLin(t,x)
% Nonlinear Ordinary Differential Equation
% x(1) = Position
% x(2) = Rate
xdot = [x(2)
-10*x(1) + 0.8*x(1)^3 - x(2)];
16
Linear and Stiffening
Cubic Springs: Small and
Large Initial Conditions
Cubic term
17
Linear and nonlinear responses are indistinguishable with small initial condition
Linear and Weakening Cubic Springs:
Small and Large Initial Conditions
Cubic term
18
Linear, Time-Varying (LTV)
Approximation of
Perturbation Dynamics!
19
Stiffening Linear-Cubic
Spring Example
Nonlinear, time-invariant (NTI) equation
x!1 (t) = f1 = x2 (t)
x!2 (t) = f2 = !10x1 (t) ! 10x13 (t) ! x2 (t)
Integrate equations to produce nominal path
! x (0) $
# 1N
&'
# x2 (0) &
" N
%
! f
1
( ## f N
2N
0 "
tf
$
& dt '
&
%
! x (t) $
# 1N
& in !0,t $
" f%
# x2 (t) &
" N
%
Analytical evaluation of partial derivatives
! f1
! f1
! f1
! f1
= 0;
=1
= 0;
=0
! x1
! x2
!u
!w
! f2
! f2
= "10 " 30x1N 2 (t);
= "1
! x1
! x2
! f2
! f2
= 0;
=0
!u
!w
20
Nominal (NTI) and Perturbation
(LTV) Dynamic Equations
Nonlinear, time-invariant (NTI) nominal equation
x! N (t) = f[x N (t)], x N (0) given
!!!!!!!!!!!!!!!!
x!1N (t) = x2 N (t)
x!2 N (t) = !10x1N (t) ! 10x1N 3 (t) ! x2 N (t)
Example
! x (0) $ !
$
# 1N
&=# 0 &
# x2 (0) & " 9 %
" N
%
Perturbations approximated by linear, time-varying (LTV) equation
!!x(t) = F(t)!x(t), !x(0) given
""""""""""""""""
# !!x1 (t) & #
0
%
(=%
2
%$ !!x2 (t) '( %$ " 10 + 30x1N (t)
(
)
1 & # !x1 (t) &
(%
(
"1 ( % !x2 (t) (
$
'
'
Example
" !x (0) % "
%
1
$
'=$ 0 '
$# !x2 (0) '& # 1 &
21
Comparison of Approximate and Exact Solutions
Initial Conditions
x2 N (0) = 9
!x2 (0) = 1
x2 N (t)+ !x2 (t) = 10
x2 (t) = 10
x N (t)
!x(t)
x N (t)+ !x(t)
x(t)
x! N (t)
!!x(t)
x! N (t) + !!x(t)
x! (t)
22
Suppose Nominal Initial Condition
is Zero
Nominal solution remains at equilibrium
x! N (t) = f[x N (t)], x N (0) = 0, x N (t) = 0 in [ 0, ! ]
Perturbation equation is linear and time-invariant (LTI)
" !!x (t) % "
0
1 %" !x1 (t) %
1
'$
$
'=$
'
"
%
(1
(10
(
30
0
(
)
'&$# !x2 (t) '&
$# !!x2 (t) '& $# #
&
23
Separation of the!
Equations of Motion into
Longitudinal and LateralDirectional Sets!
24
Rigid-Body Equations of
Motion (Scalar Notation)
Grumman F9F
State Vector
•!
Rate of change of
Translational Velocity
•!
Rate of change of
Translational Position
u! = X / m ! g sin " + rv ! qw
v! = Y / m + g sin # cos" ! ru + pw
w! = Z / m + g cos # cos" + qu ! pv
x! I = ( cos! cos" ) u + ( # cos $ sin " + sin $ sin ! cos" ) v + ( sin $ sin " + cos $ sin ! cos" ) w
y! I = ( cos! sin " ) u + ( cos $ cos" + sin $ sin ! sin " ) v + ( # sin $ cos" + cos $ sin ! sin " ) w
z!I = ( # sin ! ) u + ( sin $ cos! ) v + ( cos $ cos! ) w
•!
Rate of change of
Angular Velocity
(Ixy = Iyz = 0)
(
{ (
(
q! = "# M ! ( I xx ! I zz ) pr ! I xz p 2 ! r 2
(
{ (
r! = I xz L + I xx N ! I xz I yy ! I xx ! I zz
•!
Rate of change of
Angular Position
)
)$% ÷ I
) r + "# I
(
)
})
(
)
(
)
})
(
)
p! = I zz L + I xz N ! I xz I yy ! I xx ! I zz p + "# I xz2 + I zz I zz ! I yy $% r q ÷ I xx I zz ! I xz2
yy
2
xz
+ I xx I xx ! I yy $% p q ÷ I xx I zz ! I xz2
!! = p + ( q sin ! + r cos ! ) tan "
"! = q cos ! # r sin !
$! = ( q sin ! + r cos ! ) sec "
!
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#"
x1 $
& !
x2 & #
& #
x3 & #
x4 & #
& #
x5 & #
x6 & #
&=#
x7 & #
x8 & ##
&
x9 & #
& #
x10 & #
x11 & #
& #"
x12 &%
u
v
w
x
y
z
p
q
r
'
(
)
$
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&%
25
Reorder the State Vector
!
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#"
x1 $
& !
x2 & #
& #
x3 & #
x4 & #
& #
x5 & #
x6 & #
&=#
x7 & #
x8 & ##
&
x9 & #
& #
x10 & #
x11 & #
& #"
x12 &%
u
v
w
x
y
z
p
q
r
'
(
)
$
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&%
First six elements of
the state are
longitudinal variables
Second six elements of
the state are lateraldirectional variables
!
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#"
x1 $
&
x2 &
x3 &
&
x4 &
&
x5 &
! x Lon
x6 &
& =#
x7 &
#" x Lat'Dir
x8 &
&
x9 &
x10 &
&
x11 &
&
x12 &
% new
!
#
#
#
#
#
#
$ #
&=#
&% #
#
#
#
#
#
#
#"
26
u
w
x
z
q
(
v
y
p
r
)
*
$
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&%
Longitudinal Equations of Motion
Dynamics of velocity, position, angular rate,
and angle primarily in the vertical plane
u! = X / m ! gsin " + rv ! qw
" x!1 = f1
w! = Z / m + g cos # cos" + qu ! pv " x!2 = f2
x! I = ( cos! cos" ) u +
( # cos $ sin" + sin $ sin! cos" ) v + (sin $ sin" + cos $ sin! cos" ) w
z!I = ( # sin ! ) u + ( sin $ cos! ) v + ( cos $ cos! ) w
q! = "# M ! ( I xx ! I zz ) pr ! I xz ( p 2 ! r 2 ) $% ÷ I yy
&! = q cos ' ! r sin '
" x! 3 = f3
" x! 4 = f4
" x!5 = f5
" x!6 = f6
x! Lon (t) = f[x Lon (t), u Lon (t), w Lon (t)]
27
Lateral-Directional
Equations of Motion
Dynamics of velocity, position, angular rate,
and angle primarily out of the vertical plane
v! = Y / m + gsin ! cos" # ru + pw
" x! 7 = f7
( # sin ! cos$ + cos ! sin" sin$ ) w
" x!8 = f8
y! I = ( cos" sin$ ) u + ( cos ! cos$ + sin ! sin " sin$ ) v +
(
{ (
)
(
})
)$% p} q ) ÷ ( I
)
p! = I zz L + I xz N ! I xz I yy ! I xx ! I zz p + "# I xz2 + I zz I zz ! I yy $% r q ÷ ( I xx I zz ! I xz2 ) " x!9 = f9
(
{ (
)
(
r! = I xz L + I xx N ! I xz I yy ! I xx ! I zz r + "# I xz2 + I xx I xx ! I yy
I ! I xz2 ) " x!10 = f10
xx zz
!! = p + ( qsin ! + r cos ! ) tan " " x!11 = f11
#! = ( qsin ! + r cos ! ) sec"
" x!12 = f12
x! LD (t) = f[x LD (t), u LD (t), w LD (t)]
28
Sensitivity to Small Motions
(12 x 12) stability matrix for the entire system
"
$
$
F(t) = $
$
$
$
#
! f1
! x1
...
! f12
! x1
! f1 %
'
! x12 '
... ... '
'
! f12 '
...
! x12 '
&
...
Four (6 x 6) blocks distinguish longitudinal and lateral-directional effects
Effects of lateral-directional
perturbations on longitudinal motion
Effects of longitudinal perturbations
on longitudinal motion
" F
Lon
F = $ Lat!Dir
$ FLon
#
Effects of longitudinal perturbations
on lateral-directional motion
Lon
%
FLat!Dir
'
FLat!Dir '
&
Effects of lateral-directional perturbations
on lateral-directional motion
29
Sensitivity to Small Control Inputs
(12 x 6) control matrix for the entire system
# ! f1
! f1
%
!" E
!" T
%
! f2
% ! f2
!" E
!" T
G(t) = %
...
...
%
% !f
! f12
12
%
!"
E
!" T
$
! f1
! f1
! f1
! f1
&
!" SF (
(
! f2
! f2
! f2
! f2
(
!" F
!" A
!" R
!" SF (
...
...
...
...
(
(
! f12
! f12
! f12
! f12
!" F
!" A
!" R
!" SF ('
!" F
!" A
!" R
Four (6 x 3) blocks distinguish longitudinal and lateraldirectional control effects
Effects of longitudinal controls
on longitudinal motion
" G
Lon
G = $ Lat!Dir
$ G Lon
#
Effects of longitudinal controls
on lateral-directional motion
Effects of lateral-directional
controls on longitudinal motion
%
G Lon
Lat!Dir
'
G Lat!Dir '
&
Effects of lateral-directional controls
on lateral-directional motion
30
Sensitivity to Small Disturbance Inputs
Disturbance input vector and perturbation
!
#
#
#
w(t) = #
#
#
#
#
"
uw (t) $
&
ww (t) &
qw (t) &
&
vw (t) &
&
pw (t) &
rw (t) &
%
Axial wind, m / s
Normal wind, m / s
Pitching wind shear, deg / s or rad / s
Lateral wind, m / s
Rolling wind shear, deg / s or rad / s
Yawing wind shear, deg / s or rad / s
"
$
$
$
!w(t) = $
$
$
$
$
#
!uw (t) %
'
!ww (t) '
!qw (t) '
'
!vw (t) '
'
!pw (t) '
!rw (t) '
&
Four (6 x 3) blocks distinguish longitudinal and
lateral-directional effects
Effects of longitudinal disturbances
on longitudinal motion
" L
Lon
L = $ Lat!Dir
$ L Lon
#
Effects of longitudinal disturbances
on lateral-directional motion
Effects of lateral-directional
disturbances on longitudinal motion
%
LLon
Lat!Dir
'
L Lat!Dir '
&
Effects of lateral-directional disturbances 31
on lateral-directional motion
Decoupling Approximation
for Small Perturbations
from Steady, Level Flight!
32
Restrict the Nominal Flight Path
to the Vertical Plane
Nominal State Vector
Nominal lateral-directional motions are zero
!
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#"
x! Lat ! DirN = 0
x Lat ! DirN = 0
Nominal longitudinal equations reduce to
u! N = X / m ! gsin " N ! qN wN
w! N = Z / m + g cos" N + qN u N
x! I N = ( cos" N ) u N + ( sin " N ) wN
z!I N = ( ! sin " N ) u N + ( cos" N ) wN
q! N =
M
I yy
"!N = qN
x1 $
&
x2 &
x3 &
&
x4 &
&
x5 &
! x Lon
x6 &
& =#
x7 &
#" x Lat'Dir
x8 &
&
x9 &
x10 &
&
x11 &
&
x12 &
%N
$
&
&% N
uN $
&
wN &
xN &
&
zN &
&
qN &
(N &
&
0 &
0 &
0 &
&
0 &
0 &
0 &%
!
#
#
#
#
#
#
#
=#
#
#
#
#
#
#
#
#
"
33
Restrict the Nominal Flight Path to
Steady, Level Flight
•!
Specify nominal airspeed (VN) and altitude (hN = –zN)
•!
Calculate conditions for trimmed (equilibrium) flight
–! See Flight Dynamics and FLIGHT program for a
solution method
Trimmed State Vector
is constant
0 = X / m ! gsin " N ! qN wN
0 = Z / m + g cos" N + qN u N
VN = ( cos" N ) u N + ( sin " N ) wN
0 = ( ! sin " N ) u N + ( cos" N ) wN
0=
M
I yy
0 = qN
"
$
$
$
$
$
$
$#
"
uTrim
%
$
'
wTrim
$
'
$ V t (t
'
0)
$ N(
' =$
zN
'
$
'
0
$
'
! &Trim $ !
Trim
#
u
w
x
z
q
%
'
'
'
'
'
'
'
'
&
34
Small Longitudinal and LateralDirectional Perturbation Effects
•! Assume the airplane is symmetric and its
nominal path is steady, level flight
–! Small longitudinal and lateral-directional
perturbations are approximately uncoupled from
each other
–! (12 x 12) system is
•! block diagonal
•! constant, i.e., linear, time-invariant (LTI)
•! decoupled into two separate (6 x 6) systems
" FLon
F=$
$# 0
0
FLat!Dir
%
'
'&
" G Lon
G=$
$# 0
0
G Lat!Dir
%
'
'&
" L Lon
L=$
$# 0
0
L Lat!Dir
%
'
'&
35
(6 x 6) LTI Longitudinal
Perturbation Model
Dynamic Equation
!!x Lon (t) = FLon !x Lon (t) + G Lon !u Lon (t) + L Lon !w Lon (t)
State Vector
!x Lon
"
$
$
$
=$
$
$
$
$
#
!x1 %
" !u %
'
'
$
!x2 '
!w '
$
!x3 '
' = $ !x '
$ !z '
!x4 '
'
$
'
$ !q '
!x5 '
$ !( '
&
#
!x6 '
& Lon
Control
Vector
!u Lon
# !" T &
= %% !" E ((
%$ !" F ('
Disturbance
Vector
!w Lon
" !uwind
$
= $ !wwind
$ !q
wind
#
%
'
'
'
&
36
LTI Longitudinal
Response to Initial Pitch Rate
37
(6 x 6) LTI Lateral-Directional
Perturbation Model
Dynamic Equation
!!x Lat " Dir (t) = FLat " Dir !x Lat " Dir (t) + G Lat " Dir !u Lat " Dir (t) + L Lat " Dir !w Lat " Dir (t)
State Vector
!x Lat " Dir
#
%
%
%
=%
%
%
%
%
$
!x1 &
# !v
(
%
!x2 (
% !y
(
% !p
!x3
(
=%
!x4 (
% !r
(
% !)
!x5 (
%
!*
!x6 (
' Lat " Dir %$
&
(
(
(
(
(
(
(
('
Control
Vector
!u Lat " Dir
$ !# A '
= && !# R ))
&% !# SF )(
Disturbance
Vector
!w Lon
" !vwind
$
= $ !pwind
$ !r
wind
#
38
%
'
'
'
&
LTI Lateral-Directional Response
to Initial Yaw Rate
39
Next Time:!
Longitudinal Dynamics!
Reading:!
Flight Dynamics!
452-464, 482-486!
Airplane Stability and Control!
Chapter 7!
Learning Objectives
•!
•!
•!
•!
6th-order -> 4th-order -> hybrid equations
Dynamic stability derivatives
Long-period (phugoid) mode
Short-period mode
40
Supplemental
Material
41
How Do We Calculate the Partial
Derivatives?
F(t) =
!f
!x
x = x N (t )
u= u N (t )
w = w N (t )
G(t) =
!f
! u xu== xuNN (t(t ))
w = w N (t )
L(t) =
!f
!w
x = x N (t )
u= u N (t )
w = w N (t )
•! Numerically
–! First differences in f(x,u,w)
•! Analytically
–! Symbolic evaluation of analytical
models of F, G, and L
42
Numerical Estimation of the
Jacobian Matrix, F(t)
! f1
(t ) "
!x1
! f2
(t ) "
!x1
$ ( x + #x )
1
& 1
x2
&
f1 &
!
&
xn
&%
'
$ ( x * #x )
1
)
& 1
x2
)
&
) * f1 &
!
)
&
xn
)(
&%
2#x1
$ ( x + #x )
1
& 1
x2
&
f2 &
!
&
xn
&%
'
$ ( x * #x )
1
)
& 1
x2
)
&
) * f2 &
!
)
&
xn
)(
&%
2#x1
'
)
)
)
)
)( x=x N (t )
u=u N (t )
w=w N (t )
;
'
)
)
)
)
)( x=x N (t )
u=u N (t )
w=w N (t )
;
! f1
(t ) "
!x2
! f2
(t ) "
!x2
$
x1
&
& ( x + #x2 )
f1 & 2
!
&
xn
&%
'
$
x1
)
&
)
& ( x2 * #x2 )
) * f1 &
!
)
&
xn
)(
&%
'
)
)
)
)
)( x=x N (t )
u=u N (t )
w=w N (t )
2#x2
$
x1
&
& ( x + #x2 )
f2 & 2
!
&
xn
&%
'
$
x1
)
&
)
& ( x2 * #x2 )
) * f2 &
!
)
&
xn
)(
&%
'
)
)
)
)
)( x=x N (t )
u=u N (t )
w=w N (t )
2#x2
Continue for all n x n elements of F(t)
43
Numerical Estimation of the
Jacobian Matrix, G(t)
! f1
(t ) "
!u1
! f2
(t ) "
!u1
$ ( u + #u )
1
& 1
u2
&
f1 &
!
&
um
&%
'
$ ( u * #u )
1
)
& 1
u2
)
&
) * f1 &
!
)
&
um
)(
&%
2#u1
$ ( u + #u )
1
& 1
u2
&
f2 &
!
&
um
&%
'
$ ( u * #u )
1
)
& 1
u2
)
&
) * f2 &
!
)
&
um
)(
&%
2#u1
'
)
)
)
)
)( x=x N (t )
u=u N (t )
w=w N (t )
'
)
)
)
)
)( x=x N (t )
u=u N (t )
;
w=w N (t )
;
! f1
(t ) "
!u2
! f2
(t ) "
!u2
$
u1
&
& ( u + #u2 )
f1 & 2
!
&
um
&%
'
$
u1
)
&
)
& ( u2 * #u2 )
) * f1 &
!
)
&
um
)(
&%
'
)
)
)
)
)( x=x N (t )
u=u N (t )
w=w N (t )
2#u2
$
u1
&
& ( u + #u2 )
f2 & 2
!
&
um
&%
'
$
u1
)
&
)
& ( u2 * #u2 )
) * f2 &
!
)
&
um
)(
&%
'
)
)
)
)
)( x=x N (t )
u=u N (t )
2#u2
Continue for all n x m elements of G(t)
44
w=w N (t )
Numerical Estimation of the
Jacobian Matrix, L(t)
! f1
(t ) "
!w1
! f2
(t ) "
!w1
$ ( w + #w )
1
1
&
w2
&
f1 &
!
&
ws
&%
'
$ ( w * #w )
1
1
)
&
w2
)
&
) * f1 &
!
)
&
ws
)(
&%
2#w1
$ ( w + #w )
1
1
&
w2
&
f2 &
!
&
ws
&%
'
$ ( w * #w )
1
1
)
&
w2
)
&
) * f2 &
!
)
&
ws
)(
&%
2#w1
'
)
)
)
)
)( x=x N (t )
u=u N (t )
w=w N (t )
'
)
)
)
)
)( x=x N (t )
u=u N (t )
;
w=w N (t )
;
! f1
(t ) "
!w2
! f2
(t ) "
!w2
$
w1
&
& ( w2 + #w2 )
f1 &
!
&
ws
&%
'
$
w1
)
&
)
& ( w2 * #w2 )
) * f1 &
!
)
&
ws
)(
&%
2#w2
$
w1
&
& ( w2 + #w2 )
f2 &
!
&
ws
&%
'
$
w1
)
&
)
& ( w2 * #w2 )
) * f2 &
!
)
&
ws
)(
&%
2#w2
Continue for all n x s elements of L(t)
45
'
)
)
)
)
)( x=x N (t )
u=u N (t )
w=w N (t )
'
)
)
)
)
)( x=x N (t )
u=u N (t )
w=w N (t )