Page 155 - Dynamics of Mechanical Systems

0593_C05_fm Page 136 Monday, May 6, 2002 2:15 PM

136 Dynamics of Mechanical Systems

v P B v P B
v Q v Q

P Q P Q O

FIGURE 5.4.5 FIGURE 5.4.6
A body B with particles P and Q having parallel Location of the center of zero velocity for a body
velocities with non-equal magnitudes. having distinct particles with parallel but unequal
velocities.
To summarize, we see that if a body has planar motion, there exists a unique point O
of the body (or the body extended) that has zero velocity. O may be located at the
intersection of lines through two points that are perpendicular to the velocity vectors of
the points. Alternatively, O may be located on a line perpendicular to the velocity vector
P
of a single point P of the body at a distance v /ω from P (see Figure 5.4.7). Finally,
when the zero velocity center O is located, the body may be considered to be rotating
about O. Then, the velocity of any point P of the body is proportional to the distance from
O to P and is directed parallel to the plane of motion of B and perpendicular to the line
segment OP.

5.5 Illustrative Example: A Four-Bar Linkage
Consider the planar linkage shown in Figure 5.5.1. It consists of three links, or bars (B ,
1
B , and B ), and four joints (O, P, Q, and R). Joints O and R are fixed while joints P and Q
2
3
may move in the plane of the linkage. The ends of each bar are connected to a joint; thus,
the bars may be identified (or labeled) by their joint ends. That is, B is OP, B is PQ, B is
1
3
2
QR. In this context, we may also imagine a fourth bar B connecting the fixed joints O and
4
R. The system then has four bars and is thus referred to as a four-bar linkage.
The four-bar linkage may be used to model many physical systems employed in mech-
anisms and machines, particularly cranks and connecting rods. The four-bar linkage is
thus an excellent practical example for illustrating the concepts of the foregoing sections.
In this context, observe that bars OP(B ) and RQ(B ) undergo pure rotation, while bar
1
3
PQ(B ) undergoes general plane motion, and bar OR(B ) is fixed, or at rest.
4
2
v P
B
Q
B 2
P P
P
v
| | / ω
B 3
B 1
R
O O
FIGURE 5.4.7 FIGURE 5.5.1
Location of the center for zero velocity knowing A four-bar linkage.
the velocity of one particle and the angular
speed.