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Appendix I / Utilities / Point Mass Kinematics
_ _
22.5.3 Average Angular Rates from Direction Cosines
AVG_W_RATES takes the transform defining the orientation of frame (B)
with respect to frame (A) at time (t - ∆t) and again at time (t), and returns
the average angular rate over time interval (∆t).
AVG_W_RATES
≡
ϕ
AR
The transformation over time (∆t) is,
∆ T
22.5-2
B
B B
( T ( t − ∆t
) , T , ω , ∆t
)
B
B
( T ( t − ∆t
) , T )
A
A
B
B
( t ) : = T ( t − ∆t
) ⋅ T ( t )
A
A
A
A
A,
B
Equation 22.5-9
Equation 22.5-10
( ( ) ( ) ( ) ( ) ( ) ( ) ) T
∆T
6 − ∆T
6 , ∆T
7 − ∆T
7 , ∆T
2 − T 2
2 ⋅ Λ : =
∆
Λ
≥
10
−12 B
−1
⇒ Λ ⋅ ∆t
⋅ ωA,
B : = Λ ⋅ sin
−12
B
A, B
Λ < 10 ⇒ ω =
22.5.4 Dynamics of a Point in Inertial Space
0
3
Equation 22.5-11
( Λ )
Equation 22.5-12
Equation 22.5-13
Consider a general point (p) moving with respect to frame (B) that is itself
in motion with respect to the non-rotating frame (A). PTM_DYNAMICS
provides the inertial velocity and acceleration of point (p), expressed in (B).
PTM_DYNAMICS
B B B B B B B B B
( Aa,
b , Va,
b , ω&
A,
B , ωA,
B , Pb,
p , P&
b, p , &P
&
b, p , k , Aa,
p , Va,
p )
If (p) is moving with respect to (B) then (k = 0) and,
V : = V + P&
+ ω ×
a, p
a, b
b,
p
A,
B
P
b,
p
Equation 22.5-14
Equation 22.5-15