Page 89 - Dynamics of Mechanical Systems

0593_C03_fm Page 70 Monday, May 6, 2002 2:03 PM

70 Dynamics of Mechanical Systems

Observe that, if r is a constant, P moves on a circle and Eq. (3.8.6) has the same form as
Eq. (3.7.6). Similarly, when r is a constant Eq.s (3.8.7) and (3.7.7) have identical forms.
Finally, it might be instructive to note that when r is not a constant and when dθ/dt is
not zero, the last term in Eq. (3.8.7) is not zero. This term, called the Coriolis acceleration,
often produces anti-intuitive results in the analysis of problems with moving bodies. We
will discuss the kinematics of rigid bodies in the next chapter.

Example 3.8.1: Motion of a Cam Follower
Let a cam have a profile given by the equation:

r =+ sinθ a b (3.8.9)
>
a b
Let P be a “follower” of the cam profile as depicted in Figure 3.8.2. If the cam is fixed and
θ
˙
if the follower moves along the profile so that the angular rate is constant, determine
the radial n and transverse n components of the velocity and acceleration of P.
θ
r
Solution: From Eq. (3.8.3), the velocity of P is:
P
V = ˙ n r + rθ ˙ n (3.8.10)
r θ
Then by substituting from Eq. (3.8.9) we obtain:

)
)
+
V = ( cosθθ ˙ n +( a b sinθθ ˙ n θ (3.8.11)
P
b
r
Similarly, from Eq. (3.8.4) the acceleration of P is:
a = ( rrθ ˙ 2 n ) r ( θ 2 r ˙ n ) θ (3.8.12)
+
r + ˙θ
˙˙
−
P
˙˙
Then, by substituting from Eq. (3.8.9) we obtain:
)
a (
)
a =− − 2 sinθθ 2 ˙ n +(2 cosθθ 2 ˙ n θ (3.8.13)
P
b
b
r
Y n θ n r

P
a + b r

O θ
a a X
FIGURE 3.8.2
A fixed cam and follower P. a - b