longitudinal equation of motion
A.Kaviyarasu
Assistant Professor
Department of Aerospace Engineering
Madras Institute Of Technology
Chromepet, Chennai
Prepared
by
Longitudinal Equation of motion• The six aircraft equations of motion (EOM) can be decoupled into twosets of three equations. These are the three longitudinal EOM and thethree lateral directional EOM.
• This is convenient in that it requires only three equations to be solvedsimultaneously for many flight conditions.
• For example, an aircraft in wings-level flight with no sideslip and apitching motion can be analyzed using only the longitudinal EOMbecause the aircraft does not have any lateral-directional motion.
Kaviyarasu A, MIT Chennai
• The three longitudinal EOM consist of the x force, z force, and ymoment equations, namely,
0For longitudinal equation P R V .
.
.
( )
( )
x
yy A T
z
F m U WQ
U I M M
F m W QU
Kaviyarasu A, MIT Chennai
sin cos sin cos
sin cos
cos cos sin cos sin
y y
T
A T
T
Three force equation
m U QW RV mg D A L A T
m V RU PW mg F F
m W PV QU mg D A L A T
sin cos gx gzF mg F mg
cos sinFgx Fgz
mg mg
Kaviyarasu A, MIT Chennai
Component of gravity resolved into aircraft axis
Total differntiationof Fx
.
.
x x x x xx
F F F F FF dU dW dW
u w w
.
.
x x x x xx
F F F F FF U W W
u w w
( )u u because of smalldisturbance
.
.
x x x x xx
F F F F FF u w w
u w w
. .
u,w, , ,The forces in the X direction are a function of w
Kaviyarasu A, MIT Chennai
Kaviyarasu A, MIT Chennai
u U
x x x x xx
F F F F FF u w w
u w w
, , .u w etc are the changes in the parameters and as the perturbations are small
0 1 multiply and divide the st three terms byU.
.
0 0 0. .
0 0 0
x x x x xx
F u F w F w F FF U U U
u U w U Uw
0U U.
.
. .
x x x x xx
F u F w F w F FF U U U
u U w U Uw
.
`u w w
uU U U
Kaviyarasu A, MIT Chennai
same U on denominator and neumarotor xU
F mu muU
`u
uU
`xF mu mU u
. .
0. .
`x x x x xF F F F F
mU u U u U Uu
Kaviyarasu A, MIT Chennai
xu
F mu mUU
.
0. .
` ` ` `x x x x x
xaF F F F F
mU u U u U U Fu
xaF applied aerodynamics force
Divided by Sqin the above equation
.
. .
1 1 1 ` ` ` `
x x x x x xamU U F F F F mU F Fu u
Sq Sq u Sq Sq Sq Sq Sq
21 A ( )
2s wing rea q v Dynamic pressure
Kaviyarasu A, MIT Chennai
. . , ,
2
x x xF F F cmultiply and divide the terms by
U
c mean aerodynamic chord
Kaviyarasu A, MIT Chennai
.
1 1 2 1 2 cos` ` ` `
2 2
x x x x xa
Fxa
mU U F F c U F mg c U F Fu u C
Sq Sq u Sq U sq c sq U Sq c sq
cos` ` ` `2 2
xa
xu x x xq Fxa
mU c mg c Fu C u C C C C
Sq U sq U sq
.
1 1 2 1 2, , ,
x x x x
xu x x xq
U F F U F U FC C C C
Sq u Sq sq c Sq c
Kaviyarasu A, MIT Chennai
cos` ` ` `2 2
xa
xu x x w xq Fxa
mU c c Fu C u C C C C C
Sq U U sq
w
mgC
sq
Z u,w,w, ,The forces in the direction are a function of
1 multiply and divide the st three terms byU
Kaviyarasu A, MIT Chennai
z z z z zz
F F F F FF u w w
u w w
z z z z zz
F u F w F w F FF U U U
u U w U w U
` ` `z z z z z
zF F F F F
F U u U Uu w w
` ` `u w w
uU U U
`
: 1`
z z z zF F F FNote U as
w w U
( )zF m W PV QU 0P V
( )zF m W QU
( ) ( )m W U Q .
.
( )W
m U UU
`m U m U
Kaviyarasu A, MIT Chennai
` ` `u w w
uU U U
` ` ` `z z z z zF F F F F
m U m U U u Uu
SDivided by q
`` ` `
z z z z zm U m U F F F F FU u U
Sq Sq u
Kaviyarasu A, MIT Chennai
1 1 1 1`` ` `
z z z z zm U m U U F F F F Fu U U
Sq Sq Sq u Sq Sq Sq Sq
1 1 1 1`` ` `
z z z z z
Fxa
U F m U F F m U F Fu C
Sq u Sq Sq Sq Sq Sq Sq
1 1 1 1` ` `
z z z z z
Fza
U F mU F F mU F F Fzau CSq u Sq Sq Sq Sq Sq Sq Sq
` ` `2 2
Zu z z zq Fza
mU c mU c FzaC u C C C Cw Sin CSq U Sq U Sq
1 2 1 1 2 1, , , ,
z z z z z
Zu z z zq
U F U F F U F FC C C C Cw
Sq u Sq c Sq Sq c Sq
similarly for moment about yaxis
. .
. .
M M M M MM u w w
u w w
0 ( )M
there is nochange inM due tochange in
Kaviyarasu A, MIT Chennai
. .
. . ` ` `
M M M MM U u
u
..
we know that M Iy .. . .
. . ` ` `
M M M MIy U u
u
Kaviyarasu A, MIT Chennai
SAfter dividing by qc it becomes
... .
. .
1 1 1 ` ` `
Iy U M M M Mu
Sqc Sqc u Sqc Sqc Sqc
... .
. .
1 1 1 ` ` ` m
U M M M Iy Mu C
Sqc u Sqc Sqc Sqc Sqc
... .
. .
1 1 1 ` ` ` m
U M M M Iy Mu C
Sqc u Sqc Sqc Sqc Sqc
... .
. .
1 1 1 ` ` ` m
U M M M Iy Mu C
Sqc u Sqc Sqc Sqc Sqc
..
` ` `2 2
mu m mq m
c Iy cC u C Cm C C
U Sqc U
Kaviyarasu A, MIT Chennai
cos` ` ` `2 2
xa
xu x x xq w Fxa
mU c c Fu C u C C C C C
Sq U U sq
` ` `2 2
zaZu z z zq Fza
FmU c mU cC u C C C Cw Sin C
Sq U Sq U Sq
..
` ` `2 2
mu m mq m
c Iy cC u C Cm C C
U Sqc U
• The aircraft is flying in straight and level flight at 40,000 ftwith a velocity of 600 ft/sec (355 knots), and thecompressibility effects will be neglected. For this aircraft thevalues are as follows
Kaviyarasu A, MIT Chennai
Kaviyarasu A, MIT Chennai
6 2
,
0
0.62
5800
600 / sec
2400
2.62 10
2 0.088
0.392
0.74
2.89
20.2
2 1.48
2 ( 1.54) 0.367 2 1.13
y
xu D
Dx L
w L
t
zu L
t
m Lz
t
Mach
m slugs
U ft
S sq ft
I slug ft
C C
CC C
mgC C
Sq
l
c
c ft
C C
dC CC
di
C
,
4.42 0.04 4.46
2 2.56 1.54 3.94
Lz D
t
mzq
t
CC
dCC K
di
Kaviyarasu A, MIT Chennai
,
,
2
2
2 1.54 0.367 2 2.89 3.27
0.14 4.42 0.619
2 2.56 1.54 2.89 11.4
0.000585 600105.1 /
2 2
5800 60013.78se
2400 105.1
t
mm
a
Lm
t
mmq
dC ltC
di c
CC SM
dC ltC K
di c
q U lb sq ft
mU
Sq
6
2
c
20.2 1.130.019sec
2 2 600
0.0168 3.94 0.066sec2
0.0168 3.27 0.0552sec2
0.0168 11.4 0.192sec2
2.62 100.514sec
2400 105.1 20.2
z
zq
m
mq
y
cC
U
cC
U
cC
U
cC
U
I
Sqc
Kaviyarasu A, MIT Chennai
cos` ` ` `2 2
xa
xu x x xq w Fxa
mU c c Fu C u C C C C C
Sq U U sq
` ` `2 2
zaZu z z zq Fza
FmU c mU cC u C C C Cw Sin C
Sq U Sq U Sq
..
` ` `2 2
mu m mq m
c Iy cC u C Cm C C
U Sqc U
13.78 0.088 0.392 0.74 0` `s u s s s
1.48 13.78 4.46 13.78 0` `u s s s s s
20 0.0552 0.619 0.514 0.192 0`s s s s s
Kaviyarasu A, MIT Chennai
2
13.78 0.088 0.392 0.74
1.48 13.78 4.46 13.78 0
0 0.0552 0.619 0.514 0.192
s
s s
s s s
4 3 297.5 79 128.9 0.998 0.677 0s s s s
4 3 20.811 1.32 0.0102 0.00695 0s s s s
4Dividing by the s coefficient
2 20.00466 0.0053 0.806 1.311 0s s s s
2 2 22 0.806 1.311s ss s s s 2 2 22 0.00466 0.0053p ps s s s
0.073 / sec
0.032
p rad
p
2 1.145 / sec
0.352
ns rad
s