Environmental Engineering Reference
In-Depth Information
w
B
C
∆q
e i
A
sin f
A
C
B
e i
into page
1
e i
f
f
O
O
Illustrating the change in direction of a basis vector e i .
Figure 8.2
more general and compute the time rate of change of any vector in terms of its
components in the rotating frame. If we consider a general vector:
W i e i
W
=
W i e i =
(8.14)
then it follows that
d W
d t
d W i
d t
=
e i
d W i
d t
d e i
d t
e i +
W i
=
d W i
d t
e i +
W i ω ×
e i .
=
(8.15)
ω ×
The second term on the right hand side is equal to
W and hence we write
d W
d t
rot + ω ×
d W
d t
=
W .
(8.16)
It should be clear why we chose to write the first term on the right hand side as we
have done for it is the time rate of change of the vector W as determined in the
rotating frame. The second term on the right hand side determines how the vector
W is carried around by the rotation. We can now use this expression to determine
the velocity and acceleration of our particle. For the velocity we get
d r
d t
d r
d t =
rot + ω ×
r ,
(8.17)
Search WWH ::




Custom Search