Page 146 - Dynamics of Mechanical Systems

0593_C05_fm Page 127 Monday, May 6, 2002 2:15 PM

Planar Motion of Rigid Bodies — Methods of Analysis 127

If two particles are moving freely in three dimensions, then each particle will have three
degrees of freedom so that the system of two particles will have six degrees of freedom.
Thus, for a system of N freely moving particles (in three dimensions), there will be 3N
degrees of freedom.
Suppose a system of two particles, moving in three dimensions, is restricted by the
requirement that the particles remain a fixed distance d from each other. Let the particles
be called P and P and let them have coordinates (x , y , z ) and (x , y , z ) and position
2
2
1
2
1
1
2
1
vectors p and p as in Figure 5.2.4. Then, the fixed-distance constraint may be expressed as:
1
2
1 (
2
p − p = d or p − ) = d 2
p
1 2 2
or as:
x − ) +( y − ) +( z − ) = d 2 (5.2.5)
1 (
2
2
2
z
y
x
2
2
1
2
1
Because this constraint may be expressed by a single equation (any one of Eqs. (5.2.5)),
the system of particles has (6 – 1), or five, degrees of freedom.
A rigid body may be considered to be a system of particles whose distances are fixed
relative to each other — such as a sandstone (see Figure 5.2.5). Suppose a body is consid-
ered to have N particles p (i = 1,…, N). Then, the fixed distances of the first three particles
i
relative to each other may be expressed by the equations:
1 ( 2 2 2 ( 2 2 3 ( 2 2
p
p
p
p − ) = a , p − ) = b , p − ) = c (5.2.6)
1
2
3
where a, b, and c are constants and p locates P relative to a fixed point (or origin) O. If
i
i
P , P , and P are not colinear, the remaining particles P (i = 4,…, N) will remain a fixed
2
i
3
1
distance from P , P , and P and from each other if the following expressions are satisfied:
2
3
1
i (
2
2
2
p − ) = a i ( , p − ) = b 2 i ( , p − ) = c i ( i = 4, …, N) (5.2.7)
2
2
p
p
p
3
i
i
1
2
where a , b , and c are constants.
i
i
i
Z
n P (x , y , z )
1 1
1
1
z Z P
P 2 B
d 1
P
1
p
1
p i
P (x , y , z ) P i
2 2
2
2
P
O 2 P
Y n
O
Y
n
y
X n x X
FIGURE 5.2.4 FIGURE 5.2.5
Two particles separated by a fixed distance. A rigid body considered a set of particles.