Thesis - Leigh Moody.pdf - Bad Request - Cranfield University

Appendix D / Point Mass Dynamics
_ _
Expanding the exponential function as an infinite series,
T
R
I
⎛
⎞
( + ∆ ) ⋅ ( ) = + ∑ ⎜ ⋅ ( − [ ω × ] ) ⋅ ∆ ⎟
⎝
⎠
∞
I
1
R n
n
t t TR
t : I 3
I,
R t
n!
17-6
n : = 1
Equation 17.3-3
Collecting even and odd powers, and expressing them as power series
expansions of sine and cosine functions,
I
3
−
ω
−1
I,
R
⋅
T
I
( t + ∆t
) ⋅ T ( t )
R
−2
R 2
[ ω × ] ⋅ sin ( ω ⋅ ∆t
) + ω ⋅ [ ω × ] ⋅ 1 − cos ( ω ⋅ ∆t
)
I,
R
R
I
I,
R
R
I,
R
: =
I,
R
Expanding and equating the appropriate direction cosines,
T
R
I
I
R I
( t + ∆t
) ⋅ T ( t ) − T ( t ) ⋅ T ( t + ∆t
)
− 2 ⋅ ω
−1
I,
R
R
R [ ω × ] ⋅ sin ( ω ⋅ ∆t
)
I,
R
I
R
I,
R
( )
I,
R
Equation 17.3-4
: =
Equation 17.3-5
The average angular rate of the rotating frame with respect to the reference
axes over the time interval (∆t) is therefore,
ω
R
I, R
: =
R
ΛI
R
I
−1
⋅ sin
Λ ⋅ ∆t
R ( Λ )
I
Equation 17.3-6
( ( ) ( ) ( ) ( ) ( ) ( ) ) T
R
I
R
I
R
I
∆T
6 − T 8 , ∆T
7 − T 3 , ∆T
2 − T 4
I
17.4 Inertial Velocity Vector
R
I
2 ⋅ Λ
The inertial velocity of point (b) with respect to point (a) is,
V
I
b
: =
I
R
R
I
: =
R
T&
⋅ P + T ⋅ P&
R
a,
b
I
R
R
a,
b
Substituting for the transformation matrix derivatives,
I
R
Equation 17.3-7
Equation 17.4-1