Page 131 - Dynamics of Mechanical Systems

0593_C04*_fm Page 112 Monday, May 6, 2002 2:06 PM

112 Dynamics of Mechanical Systems

From Figure 4.13.2, however, it is seen that b may be expressed in terms of a, r, and θ as:

b =− cosθ (4.13.11)
a r
By substituting from Eq. (4.13.11) into (4.13.10), we can solve for the angular speed ratio
Ω/ω as:

+
−
Ωω =− ( [ r 1 sin θ + cos θ)] (a rcos θ) (4.13.12)
Recall that our objective is to determine the relationship between a, r, and θ so that there
will be pure rolling between B and S at C . We have not yet used the pure rolling criterion.
1
To invoke the criterion, we recall from Eq. (4.11.2) that to have pure rolling at C , we must
1
have:
B S
ωω⋅n ⊥ = 0 (4.13.13)
S
B
where ω is the angular velocity of the shaft S relative to the ball B. From Eqs. (4.7.5),
B
S
(4.13.1), and (4.13.6), ω is:
ω
ω
ω
ω
S
B
S
B ωω = R ωω − R ωω = Ωn − n − n = − n +(Ω − )n (4.13.14)
2 1 3 1 2
From Figure 4.13.2, we see that n may be expressed in terms of n and n as:
⊥
1
2
n =−cosθ n + sinθ n 2 (4.13.15)
⊥
1
Hence, the pure rolling criterion of Eq. (4.13.13) becomes:
θ
ωcos +(Ω − ) θ (4.13.16)
ω sin = 0
By solving Eq. (4.13.16) for the angular speed ratio Ω/ω, we obtain:
Ωω = (sin θ − cos θ) sin θ (4.13.17)
Finally, by equating expressions for Ω/ω of Eqs. (4.13.12) and (4.13.17), we obtain an
equation involving only a, r, and θ. By solving this equation for a/r we obtain:

+
ar = (1 sinθ ) (cosθ − sinθ ) (4.13.18)
This is the desired geometrical relation that will ensure pure rolling at C .
1
This result can also be obtained by using a rather insightful geometrical argument:
Specifically, to have pure rolling at C , ball B may be considered as being part of an
1
imaginary cone with axis along C C and rolling on the conical end of S. This is depicted
3
2
in Figure 4.13.3, where O is at the apex of the rolling cones. Let h measure the elevation
of C above O, as shown. Then, from the figure, we see that b and h are:
1
b =− cosθ = htanθ (4.13.19)
a r