Lecture #12 Topics
Develop Reference Set and Perturbation Set
• Define new vehicle fixed frame (Stability Axes
Frame)
• Develop derivatives of forces and moments needed
in the perturbation set
• Trim
Fuselage Reference Frame
Relation between forces expressed in
vehicle-axes and aerodynamic forces in
the wind axes:
𝐹𝐴𝑋 = 𝐶𝑋 𝑞∞ 𝑆 = −𝐷𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 − 𝑌𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽 + 𝐿𝑠𝑖𝑛𝛼
𝐹𝐴𝑌 = 𝐶𝑌 𝑞∞ 𝑆 = −𝐷𝑠𝑖𝑛𝛽 + 𝑌𝑐𝑜𝑠𝛽
𝐹𝐴𝑍 = 𝐶𝑍 𝑞∞ 𝑆 = −𝐷𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 − 𝑌𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽 − 𝐿𝑐𝑜𝑠𝛼
Here the angles are the
total (reference plus the
perturbation angle)
With the above definition, determination of the perturbation force 𝑓𝐴𝑋 will require
derivatives of the following form:
𝜕𝐹𝐴𝑋
𝜕𝑈
=
𝜕𝐹𝐴𝑋 𝜕𝑉∞
𝜕𝑉∞ 𝜕𝑈
+
𝜕𝐹𝐴𝑋 𝜕𝛼
𝜕𝛼 𝜕𝑈
+
𝜕𝐹𝐴𝑋 𝜕𝛽
𝜕𝛽 𝜕𝑈
Messy derivative! So for perturbation analysis a different vehicle fixed frame is used
Stability Axes Frame
No more W
component of
velocity
F – fuselage reference frame (vehicle fixed frame – original V frame)
S – stability axes frame (new vehicle fixed frame) – fixed to the specific reference condition
Subscript 0 – used to denote the forces and moments at the reference condition
Benefit: Equations of motion do not change and simplifies perturbation force expressions!
Thus relation between reference forces
expressed in new vehicle-axes frame and
aerodynamic forces in the wind axes
frame:
𝐹𝐴𝑋0 = 𝐶𝑋0 𝑞∞ 𝑆 = −𝐷0 𝑐𝑜𝑠𝛽0 − 𝑌0 𝑠𝑖𝑛𝛽0
𝐹𝐴𝑌0 = 𝐶𝑌0 𝑞∞ 𝑆 = −𝐷0 𝑠𝑖𝑛𝛽0 − 𝑌0 𝑐𝑜𝑠𝛽0
𝐹𝐴𝑍0 = 𝐶𝑍0 𝑞∞ 𝑆 = −𝐿0
Reference velocity 𝑉∞0 = 𝑈0 𝑖𝑆 + 𝑉0 𝑗𝑆
Now, if there are perturbations
Total Velocity 𝑉∞ = (𝑈0 +𝑢)𝑖𝑆 + (𝑉0 +𝑣)𝑗𝑆 +( 𝑤)𝑘𝑆
𝑤
𝑤
~
𝑈0
0 +𝑢
Perturbation 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑎𝑡𝑡𝑎𝑐𝑘 𝑡𝑎𝑛𝛼 = 𝑈
𝑇𝑜𝑡𝑎𝑙 𝑠𝑖𝑑𝑒 𝑠𝑙𝑖𝑝 tan(𝛽0 + 𝛽) =
𝑉0 + 𝑣 𝑉0 + 𝑣
~
𝑈0 + 𝑢
𝑈0
Now, relation between forces expressed
in new vehicle-axes frame and
aerodynamic forces in the wind axes
frame becomes:
𝑇𝑊,𝑆
cos(𝛽0 + 𝛽)
= −𝑠𝑖𝑛(𝛽0 + 𝛽)
0
𝑠𝑖𝑛(𝛽0 + 𝛽) 0 𝑐𝑜𝑠𝛼
0
cos(𝛽0 + 𝛽) 0
0
1 −𝑠𝑖𝑛𝛼
𝐹𝐴 = 𝐹𝐴𝑋 𝑖𝑆 + 𝐹𝐴𝑌 𝑗𝑆 + 𝐹𝐴𝑍 𝑘𝑆
𝐹𝐴𝑋 = 𝐶𝑋 𝑞∞ 𝑆
= −𝐷𝑐𝑜𝑠𝛼𝑐𝑜𝑠(𝛽0 + 𝛽) − 𝑌𝑐𝑜𝑠𝛼𝑠𝑖𝑛(𝛽0 + 𝛽) + 𝐿𝑠𝑖𝑛𝛼
𝐹𝐴𝑌 = 𝐶𝑌 𝑞∞ 𝑆 = −𝐷𝑠𝑖𝑛(𝛽0 + 𝛽) + +𝑌𝑐𝑜𝑠(𝛽0 + 𝛽)
0
1
0
𝑠𝑖𝑛𝛼
0
𝑐𝑜𝑠𝛼
Benefit:
Forces now in
terms of
perturbation
quantities!
𝐹𝐴𝑌 = 𝐶𝑍 𝑞∞ 𝑆 = −𝐷𝑠𝑖𝑛𝛼𝑐𝑜𝑠(𝛽0 + 𝛽) − 𝑌𝑠𝑖𝑛𝛼𝑠𝑖𝑛(𝛽0 + 𝛽) − 𝐿𝑐𝑜𝑠𝛼
Look at the derivative with respect to
perturbation velocity again
𝐹𝐴𝑋 = 𝐶𝑋 𝑞∞ 𝑆
= −𝐷𝑐𝑜𝑠𝛼𝑐𝑜𝑠(𝛽0 + 𝛽) − 𝑌𝑐𝑜𝑠𝛼𝑠𝑖𝑛(𝛽0 + 𝛽) + 𝐿𝑠𝑖𝑛𝛼
𝐶𝑋 = −𝐶𝐷 𝑐𝑜𝑠𝛼𝑐𝑜𝑠(𝛽0 + 𝛽) − 𝐶𝑌 𝑐𝑜𝑠𝛼𝑠𝑖𝑛(𝛽0 + 𝛽) + 𝐶𝐿 𝑠𝑖𝑛𝛼
𝜕𝐹𝐴𝑋
|
𝜕𝑈 0
𝜕𝐶
= [ 𝜕𝑈𝑋 𝑞∞ 𝑆 +
𝜕𝑞∞
𝜕𝑈
𝐶𝑋 𝑆]|0
𝜕𝑞∞
= 𝜌∞ (𝑈0 + 𝑢)
𝜕𝑈
𝜕𝐶𝑋
= −𝐶𝐷𝑈 𝑐𝑜𝑠𝛼𝑐𝑜𝑠(𝛽0 + 𝛽) − 𝐶𝑌𝑈 𝑐𝑜𝑠𝛼𝑠𝑖𝑛(𝛽0 + 𝛽) + 𝐶𝐿𝑈 𝑠𝑖𝑛𝛼
𝜕𝑈
Done! No messy derivatives
Reference Set
How many equations do we need to
characterize the reference set?
m (U 0  Q 0W 0  V 0 R0 )  mg sin  0  FAX 0  FPX 0
m (V0  R0U 0  P0W 0 )  mg cos 0 sin  0  FAY 0  FPY 0
m (W0  P0V 0  Q 0U 0 )  mg cos 0 cos  0  FAZ 0  FPZ 0


I xx P0  I xz  R0  P0Q 0   I zz  I yy Q 0R0  LA 0  LP 0
I yy Q 0   I xx  I zz  P0 R0  I xz  P02  R02   M A 0  M P 0

Assuming 𝐼𝑥𝑦 = 𝐼𝑦𝑧 = 0

I zz R0  I xz  P0  Q 0 R0   I yy  I xx P0Q 0  N A 0  N P 0
 0  P0  Q 0 sin  0 tan  0  R0 cos  0 tan  0
 0  Q 0 cos  0  R0 sin  0
 0  Q 0 sin  0  R0 cos  0  sec 0
X 0  U 0 cos 0 cos 0  V 0  sin  0 sin  0 cos  0  cos  0 sin  0 
 W 0  cos  0 sin  0 cos  0  sin  0 sin  0 
Y0  U 0 cos 0 sin 0  V 0  sin  0 sin  0 sin  0  cos  0 cos  0 
 W 0  cos  0 sin  0 sin  0  sin  0 cos  0 
h0  U 0 sin  0  V 0  sin  0 cos 0 
 W 0  cos  0 cos 0 
How many equations do we need to
characterize the reference set? – 8
m (U 0  Q 0W 0  V 0 R0 )  mg sin  0  FAX 0  FPX 0
m (V0  R0U 0  P0W 0 )  mg cos 0 sin  0  FAY 0  FPY 0
m (W0  P0V 0  Q 0U 0 )  mg cos 0 cos  0  FAZ 0  FPZ 0


I xx P0  I xz  R0  P0Q 0   I zz  I yy Q 0R0  LA 0  LP 0
I yy Q 0   I xx  I zz  P0 R0  I xz  P02  R02   M A 0  M P 0

Assuming 𝐼𝑥𝑦 = 𝐼𝑦𝑧 = 0

I zz R0  I xz  P0  Q 0 R0   I yy  I xx P0Q 0  N A 0  N P 0
 0  P0  Q 0 sin  0 tan  0  R0 cos  0 tan  0
 0  Q 0 cos  0  R0 sin  0
 0  Q 0 sin  0  R0 cos  0  sec 0
X 0  U 0 cos 0 cos 0  V 0  sin  0 sin  0 cos  0  cos  0 sin  0 
 W 0  cos  0 sin  0 cos  0  sin  0 sin  0 
Y0  U 0 cos 0 sin 0  V 0  sin  0 sin  0 sin  0  cos  0 cos  0 
 W 0  cos  0 sin  0 sin  0  sin  0 cos  0 
h0  U 0 sin  0  V 0  sin  0 cos 0 
 W 0  cos  0 cos 0 
Total number of unknowns: 8 states 4 controls
x 0  [U 0 V 0 W 0 P0 Q 0 R0  0  0 ]
u 0  T0  E 0  A 0  R 0 
Special case: Straight equilibrium
condition
Equilibrium Condition
All state derivatives =0
U 0  V0  W0  0
P0  Q 0  R0  0
 0  0  0
Straight line flight ->
0  0
Reference Set
mg sin  0  FAX 0  FPX 0
(written in
stability axes frame)
mg cos 0 cos  0  FAZ 0  FPZ 0
mg cos 0 sin  0  FAY 0  FPY 0
Unknowns – 6 [ 0  0 T0  E 0  A 0  R 0 ]
0  LA 0  LP 0
0  M A0  MP0
0  N A0  NP0