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

Glossary
_ _
P
C
a , b
≡
XC YC ZC
( P , P , P )
a,
b
a,
b
xxxvii
a,
b
T
≡
⎛ C
P
⎜ a,
⎜
⎜ C
⎜ Pa
,
⎜
⎜ C
⎝ Pa
,
b
b
b
( 1 )
( 2 )
⎞
( ) ⎟⎟⎟⎟⎟⎟⎟ 3
Capital “P” is reserved for position vectors. To simplify the nomenclature
where a line segment starts at the origin of a Frame it is permissible to write,
C
b
P ≡
In this example the Frame (C) is by definition located at point (a).
0.15.4 Cartesian Frames and Spanning Vectors
Vectors along the axes of an orthogonal right-handed Cartesian Frame are
referred to by (X C ,Y C ,Z C ). Unit vectors in these directions are denoted by
[n C ≡ (i C ,j C ,k C )]. For Cartesian Frames (X C ,Y C ) span the basic plane,
(Y C ,Z C ) the transverse plane, and (X C ,Z C ) the longitudinal plane. The basic
plane may be referred to as the “horizontal” plane as per common
convention.
The projections of a vector V onto the planes spanned by (X,Y), (Y,Z),
(X,Z) and are denoted by V h , V r and V v respectively, where (h), (r) and (v)
refer to the projection of the vector in the “basic”, “transverse” and
“longitudinal” planes of the Frame . Hence for a position vector C
P , for a,
b
example, the following notation is used:
P
P
P
rC
a,
b
vC
a,
b
hC
a,
b
: =
: =
: =
P
C
a,
b
XC YC
T
( P , P , 0 ) ≡ basic plane vector
a,
b
a,
b
YC ZC T
( 0 , P , P ) ≡ transverse plane vector
a,
b
a,
b
XC
ZC T
( P , 0 , P ) ≡ longitudinal
plane vector
a,
b
a,
b
When dealing with the principal planes of a Frame they may be referred to
as C h , C r and C v respectively.
0.15.5 Position Vector Time Derivatives
Consider the time rate of change of a vector in an inertial Frame (I) and
rotating Frame (R). In the inertial Frame,
d
dt
I
I
I
( ) ≡ P ≡ V & P
b
b
b
⎠