longitudinal equation of motion Equation of motion •Thesixaircraftequationsofmotion(EOM)canbedecoupledintotwo setsofthreeequations.Thesearethethreelongitudinal EOMandthe threelateraldirectionalEOM.

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

Kaviyarasu A, MIT Chennai

Equilibrium and disturbed aircraft stability axes

• 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

.` `x x x x x

xF F F F F

F U u U Uu w w

Kaviyarasu A, MIT Chennai

` 1

`

x x x xF F F FU as

w w U

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

Kaviyarasu A, MIT Chennai

• 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

Thank you