Page 159 - The Mechatronics Handbook

instantaneous axis
of rotation z a
ω z a
B
r B/A
Ω
y a
r B/A ^
z o B z o k
a
r B
y a A ^
A j
v A a
^
i
a
α
r A
x a ^
k o x a
a A
O y o O ^ j y o
^ o
i o
x o x o
(a) (b)
FIGURE 9.30 General rigid body motion: (a) rigid body with translating coordinate system, (b) translating and
rotating coordinate system.
Motion of Point B Relative to O. For translating axes with no rotation, the velocity and acceleration
of point B relative to system 0 is simply, v B = v A + v B/A and a B = a A + a B/A respectively, or,

v B = v A + ω × r B/A (9.10)

a B = a A + α × r B/A + ω × ( ω × r B/A ) (9.11)

Translating and Rotating Coordinate Axes
A general way of describing the three-dimensional motion of a rigid body uses a set of axes that can
translate and rotate relative to a second set of axes, as illustrated in Fig. 9.30(b). Position vectors specify
the locations of points A and B on the body relative to x o , y o , z o , and the axes x a , y a , z a have angular
·
Ω
Ω
velocity and angular acceleration . With the position of point B given by
r B = r A + r B/A (9.12)

the velocity and acceleration are found by direct differentiation as

v B = v A + Ω × r B /A + ( v B/A ) a (9.13)

and

˙
B = a A + Ω × r B /A+ Ω ¥ ( Ω × r B /A ) + 2Ω × ( v B/A ) a + ( a B/A ) (9.14)

where (v B/A ) a and (a B/A ) a are the velocity and acceleration, respectively, of B relative to A in the x a , y a , z a
coordinate frame.
These equations are applicable to plane motion of the rigid body for which the analysis is simplified
·
·
Ω
Ω
Ω
since and have a constant direction. Note that for the three-dimensional case, must be computed
by using Eq. (9.9).
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