Page 381 - Dynamics of Mechanical Systems

0593_C11_fm Page 362 Monday, May 6, 2002 2:59 PM

362 Dynamics of Mechanical Systems

Z Y
n P(x,y,z)
z
n
θ
p(r,θ)
p
R O Y
p n
n y r

n x θ
X O X
FIGURE 11.4.1 FIGURE 11.4.2
A particle P moving in an inertial frame R. Movement of a particle P in a plane.

fixed point O in R. Let the coordinates of P be (x, y, z), the X, Y, Z Cartesian coordinates
of a point at the same position as P. Then the position vector p may be expressed as:

p = x n + y n + z n (11.4.17)
x y z
where n , n , and n are constant unit vectors parallel to the X-, Y-, and Z-axes as in Figure
y
z
x
11.4.1. The velocity of P in R is then:
v = ˙ x n + ˙ y n + ˙ z n (11.4.18)
x y z
From Eqs. (11.4.4) and (11.4.5), the partial velocities of P relative to x, y, and z are:

v = n , v = n , v = n z (11.4.19)
˙ z
˙ x
y
x
˙ y
Observe in this case that the partial velocity vectors are simply the unit vectors normally
used in analyses with Cartesian reference frames.
As a second example, consider the motion of a particle P in a plane as in Figure 11.4.2.
In this case, let the coordinates of P be the polar coordinates (r, θ) as shown. Then, the
position vector p locating P relative to the origin O is:

p = r n (11.4.20)
r

where n (and n ) are the radial (and transverse) unit vectors shown in Figure 11.4.2. The
θ
r
velocity of P is then (see Eq. (3.8.3)):
v = ˙ r n + rθ ˙ n (11.4.21)
r θ
Hence, by comparison with Eq. (11.4.5), we see that the partial velocities of P relative to
r and θ are:
v = n and v = n r θ (11.4.22)
˙ θ
˙ r
r
Observe in this case the partial velocity v is not a unit vector, neither is it dimensionless.
˙ θ
θ
˙
Instead, v has the unit of length, as a consequence of the differentiation with respect to .
˙ θ