Page 125 - Dynamics of Mechanical Systems

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106 Dynamics of Mechanical Systems

S L*
From Figure 4.10.2, a is:
a = a + ωω ×( ωω ×
S L* S O S C S C OL)
y ( [ ( z)]
=−19 .36 n + −0 .88 n z) ×− ( 0 .88 n z) × 0 .433 n + 0 .25 n (4.10.18)
y
=−19 .7 n ft sec 2
y

S
Finally, from both figures 2 ω × V is:
L
C
C
2 088 ) ×−
C
C
S
2 ωω× V L = ( − . n z ( 1 571n. y + 271n. z)
(4.10.19)
=− 2 765n. ft sec 2
x
Therefore, by substituting into Eq. (4.10.16), the acceleration of L is:
S L 2
.
.
.
a =−2 765 n − 36 79 n − 9 87 n ft sec (4.10.20)
x y z

4.11 Rolling Bodies

Rolling motion is an important special case in the kinematics of rigid bodies. It is partic-
ularly important in machine kinematics. Rolling can occur between two bodies or between
a body and a surface. Rolling between two bodies occurs when the bodies are moving
relative to each other but still are in contact with each other, with the contacting points
having zero relative velocity. Similarly, a body rolls on a surface when it is moving relative
to the surface but is still in contact with the surface with the contacting point (or points)
having zero velocity relative to the surface.
Rolling may be defined analytically as follows: Let S be a surface and let B be a body
that rolls on S as depicted in Figure 4.11.1. (S could be a portion of a body upon which B
rolls.) Let B and S be counterformal so that they are in contact at a single point. Let C be
the point of B that is in contact with S. Rolling then occurs when:

S C
V = 0 (4.11.1)

B

P

p

C

FIGURE 4.11.1
A body B rolling on a surface S.