linear flight dynamics - tu wien · 2015. 11. 27. · qsb r bb 2 cn p < 0 yaw moment –coupled...
TRANSCRIPT
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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
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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
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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
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4Automation & Control InstituteVienna University of Technology
Sign Convention
XB
YBZB
+L+P
+U
+V
+W
+C
+N
+Q
+M
+R
+N
+Y
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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
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6Automation & Control InstituteVienna University of Technology
Directional Control Deflections
Positive Rudder deflection is Trailing Edge Left (TEL) and causes
a Negative Yaw Rate (DR
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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18Automation & Control InstituteVienna University of Technology
Typical Lat-Dir Pole Plot
s
jwd
spiraldutch roll
roll
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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
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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)
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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)
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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)