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0593_C05_fm Page 150 Monday, May 6, 2002 2:15 PM

150 Dynamics of Mechanical Systems

5.8 Instant Center of Zero Acceleration
We can extend and generalize these procedures to obtain a center of zero acceleration —
that is, a point of a body (or the body extended) that has zero acceleration. To this end,
consider again a body B moving in planar motion as depicted in Figure 5.8.1. As before,
let P and Q be typical points of B, and let C be the sought-after center of zero acceleration.
Let (x , y ), (x , y ), and (x , y ) be the X–Y coordinates of P, Q, and C. Let r locate C
P P Q Q C C
relative to P. Let r have magnitude r and inclination θ relative to the X-axis as shown in
the figure. Finally, let ω and α represent the angular speed and angular acceleration of B.
Because P and C are fixed in B, their accelerations are related by the expression (see
Eq. (4.9.6)):

a = a + αα × r + ωω ×(ωω × r) (5.8.1)
P
C
Therefore, if the acceleration of C is zero, then the acceleration of P is:

a =− × − ωω ×(ωω × r) (5.8.2)
α
α
P
r
If n is a unit vector normal to the X–Y plane, then the angular velocity and angular
z
acceleration vectors may be expressed as (see Eq. (5.7.5)):
ωω = ωn and αα = αn (5.8.3)
z z
Also, from Figure 5.8.1, the position vector r may be written as:

r = r cosθ n + rsinθ n (5.8.4)
x y

Then terms α × r and ω × (ω × r) in Eq. (5.8.2) are:

αα× = −rαsin θ n + rα cos θ n (5.8.5)
r
x y
and
ωω ×( ωω × ) =−rω cos θ n − rω sin θ n (5.8.6)
2
2
r
x y
Y
n
y C
r B
P θ
α ω
P
P Q
P
C

FIGURE 5.8.1 O
X
A body B in planar motion with center
C of zero acceleration. n x