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Kinematics and dynamics of rigid bodies 33
Differentiation with respect to time is often denoted by the Newtonian dot
giving
⎡A˙
{ A ˙}
x
⎤
A˙
1 ⎢ ⎥
⎢ y ⎥
⎢A˙
⎥
⎣ z ⎦
(2.36)
If the frames used for measurement and reference differ it is necessary to
distinguish between, for example d{A} m/n /dt and {A } m/n since
d
{} A mn {} A˙
dt
/ mn /
(2.37)
In evaluating d{A} m/n /dt, we measure {A} in frame m, transform to
frame n and then differentiate {A} m/n with respect to time. The notation
{A } m/n , however, implies that {A } m is determined first and that this vector
is then transformed to frame n.
Consider the vector {A} 1/1 shown in Figure 2.12. The vector lies in the X 1 Y 1
plane of frame O 1 and rotates at a constant speed of rad/s about the
Z 1 -axis. Frame 2 has its Z-axis coincident with Oz 1 and its X-axis is coincident
with and rotates with {A} 1/1 .
If OX 1 and OX 2 were coincident at time t 0, then where A is the magnitude
of {A} 1/1 :
{A} T 1/1 [A cos t A sin t 0] (2.38)
Transforming to frame O 2 gives
{A} T 1/2 [A 0 0] (2.39)
Differentiating this with respect to time gives the following since the magnitude
A does not vary with time:
d
dt
A T {} /
12 [0 0 0]
(2.40)
Y 2
O 2
Y 1
X 2
{A} 1/1
ωt
X 1
O 1
Z 1
Z 2
Fig. 2.12
Vector rotating with constant velocity