Linear Flight Dynamics

Extra material for assignment#4

Vitaly Shaferman

27 November 2015

* Based on am EOM course and slides given at the USAF TPS

2Automation & Control InstituteVienna University of Technology

Stability and Wind Axis System

Xs

Zs,w

V

a

a

XW

YW

Vb

b

YB

XB

ZB

UV

W

Vta

b YB

XB

ZB

UV

W

Vta

b

3Automation & Control InstituteVienna University of Technology

Positive Rotation Rates and Moments

+P: Positive Roll Rate

+L: Positive Rolling Moment

Roll

Rt Wing Down

Yaw

+R: Positive Yaw Rate

+N: Positive Yawing Moment

Nose Right

4Automation & Control InstituteVienna University of Technology

Sign Convention

XB

YBZB

+L+P

+U

+V

+W

+C

+N

+Q

+M

+R

+N

+Y

5Automation & Control InstituteVienna University of Technology

Lateral Control Deflections

Positive Aileron Deflection is Trailing Edge Down (TED)

Then, a composite aileron deflection is calculated:

Positive Composite Aileron Deflection causes

a Positive Roll Rate (DP>0)

a Positive Rolling Moment (DL>0)

DDD ,, LPa

2

RL aaa

+P: Positive Roll Rate

+L: Positive Rolling

Moment

Roll

Rt Wing Down

+a

6Automation & Control InstituteVienna University of Technology

Directional Control Deflections

Positive Rudder deflection is Trailing Edge Left (TEL) and causes

a Negative Yaw Rate (DR

7Automation & Control InstituteVienna University of Technology

The Model

b a b b ot

y y a y r

t

p rqS

mVC C C

g

Va r

rlallrl

t

l

xxxx

xzrap CCrCpC

V

bC

I

Sbqr

I

Ip b b

2

rnannrn

t

n

zzzz

xzrap CCrCpC

V

bC

I

Sbqp

I

Ir b b

2

yzdmIyz

XB

ZB

8Automation & Control InstituteVienna University of Technology

Yt

YB

Vt

XBb

Yf

b a b b ot

y y a y r

t

p rqS

mVC C C

g

Va r

Side Force Equation

Cyb < 0

Cyr > 0

9Automation & Control InstituteVienna University of Technology

Roll Moment Equation (Spring, Mass, Damper)

Clr

IXX

Clp Clb Cla

Forcing Functions: Cla, Clr

Spring Constant: Clb

Damping Constant: Clp

rlallrl

t

l

xx

rap CCrCpCV

bC

I

Sbqp b b

2

10Automation & Control InstituteVienna University of Technology

-b causes larger relative wind on left wing

resulting in more lift on left wing and Cl > 0

Roll Moment - Dihedral Effect

Vt

Vt

Vn

Vn

Vt

Xs

b

Swept Wing

a

b

Yt

ab

YB

Vertical Tail

For a vertical tail above the roll

axis, +b causes –Cl

Clb < 0

rlallrl

t

l

xx

rap CCrCpCV

bC

I

Sbqp b b

2

Clb < 0

11Automation & Control InstituteVienna University of Technology

ZB

DwDw

Dv

rlallrl

t

l

xx

rap CCrCpCV

bC

I

Sbqp b b

2

YB

XB

R

Flat, skidding turn

Yt

YB

N

N

Roll Moment – Damping Terms

Clp < 0

Clr > 0

12Automation & Control InstituteVienna University of Technology

Tail above roll axis

la laDNL

DNR

rlallrl

t

l

xx

rap CCrCpCV

bC

I

Sbqp b b

2

Roll Moment – Control Terms

Cla > 0

Clr > 0

13Automation & Control InstituteVienna University of Technology

Forcing Functions: Cnr, Cna

Spring Constant: Cnb

Damping Constant: Cnr

CnbCnr

IZZ Ixy

CnrCna

Yaw Moment Equation (Spring, Mass, Damper)

rnannrn

t

n

zz

rap CCrCpCV

bC

I

Sbqr b b

2

14Automation & Control InstituteVienna University of Technology

N

Yt

YB

Vt

XBb

Yf

Yaw Moment - “Spring”

rnannrn

t

n

zz

rap CCrCpCV

bC

I

Sbqr b b

2

Cnb > 0

15Automation & Control InstituteVienna University of Technology

YB

XB

N

RDv

DYt

NtVertical Tail Contribution

YB

XB

N

R

Nw

Wing Contribution

C

C

Left Wing, Higher Lift,

Higher Drag due to Lift

rnannrn

t

n

zz

rap CCrCpCV

bC

I

Sbqr b b

2

Cnr < 0

Cnr < 0

Yaw Moment – Damping Term

16Automation & Control InstituteVienna University of Technology

rnannrn

t

n

zz

rap CCrCpCV

bC

I

Sbqr b b

2

Cnp < 0

Yaw Moment – Coupled Term

ZB

DwDw

WingWing: Positive P rotates Lift

vector forward on right wing -

decreases chord force, creates

negative Cn.

ZB

DvTail

Vertical tail above roll axis:

Positive P creates positive

Cn. Cnp > 0

17Automation & Control InstituteVienna University of Technology

YB

XB

N

DYt

r

lt

Yaw Moment – Control Terms

rnannrn

t

n

zz

rap CCrCpCV

bC

I

Sbqr b b

2

Cnr < 0

la la

Cna 0

18Automation & Control InstituteVienna University of Technology

Typical Lat-Dir Pole Plot

s

jwd

spiraldutch roll

roll

19Automation & Control InstituteVienna University of Technology

Our System

002.00

001.00k

kxu

BuAxx

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4-3

-2

-1

0

1

2

3Poles of the open and closed loop

Open Loop

Closed Loop

20Automation & Control InstituteVienna University of Technology

Original System – Dutch Roll

0 1 2 3 4 5 6 7 8 9 10-0.2

0

0.2

b (

deg)

Dutch Roll Mode - Attitudes [b,s,

s]

0 1 2 3 4 5 6 7 8 9 10-1

0

1

s (

deg)

0 1 2 3 4 5 6 7 8 9 10-0.2

0

0.2

s (

deg)

Time (s)

21Automation & Control InstituteVienna University of Technology

Original System – Roll Mode

0 1 2 3 4 5 6 7 8 9 10-0.4

-0.3

-0.2

-0.1

0p

s (

deg/s

)

Roll Mode [ps,

s]

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

s (

deg)

Time (s)

22Automation & Control InstituteVienna University of Technology

Original System – Spiral Mode

0 10 20 30 40 50 60 70 80 90 100-8

-7

-6

-5

-4x 10

-3

r s (

deg/s

)

Spiral Mode [rs,

s]

0 10 20 30 40 50 60 70 80 90 100-1.8

-1.6

-1.4

-1.2

-1

-0.8

s (

deg)

Time (s)