Page 145 - Dynamics of Mechanical Systems

0593_C05_fm Page 126 Monday, May 6, 2002 2:15 PM

126 Dynamics of Mechanical Systems

Y
C
Y Y P(x,y)
a
θ
O
X
x P(x,y) P(r,θ )
r
y
O O θ
X X
Cartesian Polar
FIGURE 5.2.1 FIGURE 5.2.2
Cartesian and polar coordinates of a point. A point restricted to move on a circle.
Using polar coordinates the constraint equation is:

r = a (5.2.2)

The number of degrees of freedom of a mechanical system is sometimes defined as the
difference between the number of coordinates needed to define the location of the particles
of the system and the number of constraint equations needed to define the restrictions on
the movement of the particles.
In three dimensions, it is generally known that an unrestrained particle has three degrees
of freedom and that an unrestrained body has six degrees of freedom (three in translation
and three in rotation). To discuss and examine this further, consider a particle P, free to
move in space as depicted in Figure 5.2.3. Then, in a Cartesian coordinate system, P may
be located by the coordinates (x, y, z) defining the distances of P from the coordinate
planes. Alternatively, P may be located by the position vector p expressed as:

p = x n + y n + z n (5.2.3)
x y z
where n , n , and n are unit vectors parallel to the coordinate axes as in Figure 5.2.3.
z
y
x
Constraints on the movement of P are often expressed in terms of the position vector
p. For example, if P is restricted to planar motion, say in the X–Y plane, then this restriction
(or constraint) may be expressed as:
⋅
pn = 0 or as z = 0 (5.2.4)
z
Z
P(x,y,z)
n z

p
O
Y
n
y
FIGURE 5.2.3 n x
A particle P moving in space. X