Download Overview Body Axis System Summary

Equations of Motion:
Force Equations
Overview
• Review
– Coordinate Transformations
• Equations of Motion: Force Equations
Body Axis System Summary
Axis
x
y
z
Translational
Velocity
Angular Velocity
U
V
W
(+ out nose)
(+ out right wing)
(+ out belly)
(+ Æ rt hand rule)
Moment (+ same)
Control Surface
P (roll rate)
Q (pitch rate)
R (yaw rate)
(+ rt wing down)
(+ pitch up)
(+ nose right)
L
M
N
Aileron (δa )
Elevator (δe )
Rudder (δr )
(+ creates a + roll)
(+ T.E. down)
x
y
(+ T.E. left)
x
y
y
z
z
z
1
Body Axis System
P L
Right Hand
Coordinate System:
x
U
Translational
Velocity:
x, y, z
U, V, W
Angular
Velocity:
QM
y
V
P, Q, R
R N
z
W
Moment:
L, M, N
Aircraft Roll (P)
Rotation about the x axis
• Ailerons
• Positive (+) right wing down
Ailerons
Aircraft Pitch (Q)
Rotation about the y axis
• Elevator
• Positive (+) nose up
Elevator
2
Aircraft Yaw (R)
Rotation about the z axis
• Rudder
• Positive (+) nose right
Rudder
6 Degrees of Freedom (DOF)
Force Eqs
y
Axis
x
Translational
Velocity
Angular Velocity
U
V
W
(+ out nose)
(+ out right wing)
(+ out belly)
P
Q
R
(+ rt wing down)
(+ pitch up)
(+ nose right)
L
M
N
(+ Æ rt hand rule)
Moment (+ same)
Control Surface
z
Moment Eqs
Aileron (δ ) Elevator (δ ) Rudder (δ )
a
e
(+ creates a + roll)
r
(+ T.E. down)
x
y
(+ T.E. left)
x
y
y
z
z
z
Newton:
∑ F =m
dV
dt
(if m=const)
I
(Valid only in inertial reference frame)
Non-Inertial (Body) Axis System:
dV
dt
=
I
B
Inertial
Acceleration
dV
dt
+ ω B × VB
B
Magnitude
Change
Direction
Change
3
=
I
Inertial
Acceleration
dV
dt
U& 
& 
V 
W& 
 B
 ΣF x 


 ΣF y  = m
 ΣF 
 z
+ ω B × VB
B
+
P
Q 
 
 R  B
+
dV
dt
U 
V 
 
W  B
U& + QW − RV 
&

V + RU − PW 
W& + PV − QU 


Linear
Centripetal
Acceleration Acceleration
4