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

Glossary
_ _
Inertial velocity is denoted by (V), acceleration by (A) and jerk by (J). The
time rate of change operator of a vector in inertial space is (DI). Similarly,
the time rate of change operator of a vector in the rotating frame is (DR).
The angular velocity of frame (r) with respect to frame (I), expressed in the
rotating frame,
ω
R
I , R
≡
( ) T
XR YR ZR
ω , ω , ω
I,
R
The relationship between these time rate of change operators on any vector,
D
I
xxxviii
I,
R
I,
R
R
( ∗ ) : = ( D + ω × ) ( ∗ )
0.15.6 Length, Scalar and Cross Products of Vectors and Unit Vectors
For this section (V) and (W) denote 3D vectors in a common Cartesian
frame (C). The length of (V) is denoted by V , or by omitting the
underline; the former being useful when taking the length of vector
expressions such as V + W . For any vector (V),
R
I,
R
( ) ( ) ( ) 2
X 2 Y 2 Z
V + V V
V : =
+
( ( ) ) ( ( ) ) ( ( ) ) 2
2
2
V 1 + V 2 V 3
V ≡
+
The length of a position vector ( C
P ),
C
a,
b
C
a,
b
a,
b
( ) ( ) ( ) 2
XC 2 YC 2 YC
P + P P
P ≡ P : =
+
a,
b
C
Pa , b
& denotes the length of the derivative of C
P a,
b , not the derivative of the
length of C
P a,
b . The inner or scalar product representing the projection of
(V) onto (W) is,
V • W
: =
∑
i:
= { X,
Y,
Z}
a,
b
a,
b
3
i i
i i
( V ⋅ W ) ≡ ( V ⋅ W )
Equivalently, the scalar product may be written as,
∑
i : = 1
[ 0 π ]
V • W : = V ⋅ W ⋅ cos ξ where ξ ∈ ,
If V and W are directed line segments (ξ) is interpreted as the angle
subtended by V and W. In particular, two vectors of non-zero length are
said to be orthogonal when their scalar product is zero. The length of a
vector expressed in terms of its scalar product is,